Til K CALCULATION OK CIIANCK-WHEKLS SCREW-CUTTING ON LATHES TUK CALCULATION OK CHANCE-WHEELS SCREW-CUTTING ON LATHES TllK CALCULATION OF CHANGE -WHEELS 1*OK SCRIM-CUTTING ON LATJIKS A PRACTICAL MANUAL I tilt Illl- USl> til- MANUFACTURERS, STUDKNTS AND LATH1CMKN 1), Dh VRIKS WIIII 4(1 ILLUSrHAriONf) K, & F, N, Si'ON, LIMITED, g; 1IAYMAUICKT $eto fortt Sl'ONT *Sr C1IAM11KRLAIN, 123 r.IHKRTY STRKKT PR Kl< ACM IP . a i unoiis < \f( uni'tirtm e lh.it the calculation ol (hani>,e- \\he<li foi jlir tuttin*; t>( diffeient pitches ol thiead on a lathe, hmu'vet simple -.itch ,i udt ulaliojp may he, is u>jnp.itativel\ but little Known, hc-int;, lor the tn.ijoii'ty ol Ihosv most ( loscly intctrstal in ttu* suhjc'ct, shioiuletl HI in>stt'i v Many \\hosr thcoK'lual knowl'<ljt' is quite 1 suliuicnl to jMKihlt tht'tn lo face the problem, have had so little jnaetual expenenre in setew-uitttng th.it tliey .u'e uiiiiblc to to <kej)ly into Ihe m.ittet, and pie.sent, in a eleui (in<l siiiiplr inatHH'i, the tlifietenl variations whiuh in.iy possibly occur. The \\n ,tlei nutubei ot ineehanics, even the ytuui;>ei ones, pu.-.tss (no sh;hl a the(*retltil knowledge to permit ol then btuhlui", up a -.ystein by themselves. thru aie, ol ionise, iiHHhaiuch who are quit< j capable o{ uoikui^', tnit tlic iH'fu.ssiuy rukulation, but so many of them I speak ftom petsonal experience regard their knowl<*d!' ( e an aunt* or less of n secret, and uy, ut any rale to theniM'lvi"., "Why should 1 impart to others what ha*', taken tut 4 M ituu'h trouble and cost me so much money lo Irani j 1 *' The pin pose ol the present treatise is to unable any one, who Is ptepared to take tine trouble to study it carefully U Irani how to *ah ulate change- wheels piopeily, vi The Calculation oj Change- Wheel*. I have deemed it expedient, foi the sake of those of my readers who have but a supeificial knowledge oi tlu- lathe, to give a shoit desenption of this tool, in so fat as it is connected with sciew-cuttmg, to winch 1 have atUeil a de&ciiption of the various types of thiuad to be met with, with the necessaiy tables appended, as also a nunihet ui piactical hints, with icfcience to anew-cutting, together with the operations connected theiewith. I have puiposcly refiained from including, 1 ; a munhei ( tables giving the change-wheels requited for the v.uiuun pitches of thicads on different lathen, in place of whuh a large numbci of piactical examples aie given which cuvei eveiy possible variation likely to be met with in practical work. Experience has taught me that the inclusitm o! such tables only leads to purely mechanical work demanding no effort of the mind, whereas, in each partieuhu c.ue, tlu consideration should be given to the special work in ham!, so that in cases of exceptional difficulty, whore one is obliged to set to woik without the assistance of such tables, the manner of calculation may not be unfamiliar, It is my earnest wish that the present work may prow useful not only to students, but also to those engaged in practical work, I), DK VRIKS. HILARY * CONTKNTS CUAITKK 1. I'AGK THF LA mi , , , .. , y CHAPTER 11, {/*) Systems ,, ,, , ,, .. no (/*) Whflt C "ha nt>e Wheels aic to he found on a Lathe . 15 {/) Thf Cutting ol Metrie Thieuds on a Lathe with Motile Leadheiew , ,. , . . , 16 (/} The Cutting of Mni'ltJih Threw Is on a Lathe with iCfij'hhh l,e,ld,seiew .. . . ,, ,, 3O if) To C*ttt Kiifjltsh Thread?, on u I'ilh< i with Metric (/) Tin C'utiini 1 , ol Metric Thiends i>n a Lathe with tyiglihh Lesidwttnv ,, ,, , 37 (,C) Thr Wheel with is? Teeth ,, ., , . ,, 30 {//} Method foi Caieulnting Appioximalo {''tnefions ., 32 {/) The I 'tool of the Sum , ,, , ,, ,. 51 {&) i\ nm up the Wheels , .. , 53 {/) Thrfc'uftmj, with Double Compound Train 5/1 (i) TJu? Cutting oi* Left-hand Threads ,, ,, ,, 55 vin The Calculation of Change- Wheels CHAPTER lit. THREADS AND THEIR CONST. RUCTION ,, Af ,, (a) Foims of Thieads 56 (b} Types of Thieads 56 (c) Screw-Cutting Tools 65 (J) Cutting the Thiead 6<; ((?) The Cutting of Double and Multiple Thi ended Sacws 73 (/) The Cutting of very Coaise Thiead 75 (g) The Hendey-Noilon System 77 THE CALCULATION OF CHANGE-WHEELS FOR SCREW-CUTTING ON LATHES owri-r- CHAPTER I THE LATHE. TliRKAHS, both internal and external, can be obtained in two different ways, the simplest of which is to cut the thread by means of taps, dies and chaseis. In the smaller sizes, the majority of internal threads aie tapped, whilst external threads are cut with dies, but in the larger sizes too much material has to be removed. Tapping, however, is far moi e general than the use of dies, as in most cases, external tin cads can be obtained in anothei way, viz. : on the lathe, whilst internal threads can only be obtained on the lathe at considerable expense, M 01 cover, internal threads are to be found in a number ol different places on the largei machine parts, and so it would be well-nigh impossible to put these pieces on the lathe foi the puipose of cutting the threads. On the othei hand, a bolt 01 screw-spindle, as a rule, can be set on the lathe, and thieads may be cut by means of a common tool. It is just for this reason that, whilst a large number of i in. external thieads are cut on the lathe, I in. threads in holes aic, with but Few exceptions, cut exclusively by tapping. The practice, however, of cutting internal threads of more than 2 in. diameter on the lathe, whenever the work-piece allows it, is becoming more and more general. The object of the present work is to give a detailed description of the way in which it is possible to cut the various B 2 The Calculation of Change-Wheeh thieads on the lathe, and thus to answer, as fully as possible, the question " How are the change-wheels to be calculated foi screw-cutting on the lathe ? " In order that this woik may also be of service to those who are not fully conversant with the lathe, the following points will be treated successively, viz the gcneial constiuction of the lathe, moie especially of those paits of the Lithe used in screw-cutting , the theory of the calculation of change-wheels and screw-cutting m practice FIG. i The lathe, as originally constructed, was not intended foi sciew-cutting Fig i shows a lathe as it was fiist constructed. On this lathe a rotary movement was imparted by means of a driving belt to the headstock and workpicce only, all other movements being executed by the operator himself. Within a compaiatively short time, however, more wa.s demanded of this machine, larger pieces were required to be machined than was possible wuh direct belt drive, and the for Screw-cuffing on Lathes, 3 double back t*vai was mtioduced , it was deshed to move the tool on the mateua.1 automatically, and to obtain this, the iest was mounted on a carriage and moved by means of a leadscrew which motion was impaitod by means of eithet a belt or a iiam of ^eais fiom the hcadstock. The intto- duction of a train of gems on the apt on made it possible not only to move the cairiage ovej the whole length of the bed foi sliding, bul also to move tlie rest automatically in a ttansverse dneetton over the eauiacie itself foi suifictm>. Finally, the* leadscrew spindle, calkd for short the " leadseicvv," wns <o arranged that by a set of gears of various diameters, a variable, but at the same time foi each train of gcnn* fixed ratio between the number of revolutions of the head- stock, I.G, the workpiecc, and the loadacrow was obtainable, thus making it possible to cut different pitches of threads on the lathe. Fig, 2 given the general arrangement of such a lathe. * B 2 4 The Colt illation of Changc-irhceh The leadscrew revolves m the leudseiew-nul, which is fixed to the apion, and, as this nut cannot revolve, it travels along the leadscrew, the caniage at the same time maKtn a coi responding movement The movement of the carnage aheady causes a ronstdeiahle piessuic on the thread of the leadsciew and the nut, whit h is still inci eased by the cutting of the tool on the matenal, and, as a natural icsult, both the leads<iew and the nut aie exposed to a ceitam amount oi wear. 'I his weai i\ ttnther mci eased by swaif and chips falling on the le.ulserew, and their getting between the nut and thiead, It is evident, as far as the leadsciew is concerned, that this wear will only affect that poition ovei which the mi! travels on the thiead. As the woik on the Kit lie v,uies HI length (but is as a inlc considerably shoitei than the m.iximwn distance between the centies), the wear of the tluead is gieatest on those parts of the leadsciew where the nut mv<*s ( and after being in use foi a certain time, it is impossible to pi event the leadsciew being scarcely worn at all at the end but considerably woi n in the centre, and worn most f nil close to the headstock, The wear of the nut, however, is fairly even. The nut was foimeily made solid, consequently it was impossible to icpair the wear. It wus soon seen, however, that it was piefeiablc to have half nuts, .so that not only can it now be repaired, but, by means of the lever a (Fig, 2), it can also be opened and closed. This has led to the attainment of a number of advantages First and foiemost, the possibility of repairing the nut just referred to A downward pressure of the level a keeps both halves of the nut closed so as to grip the losukserew. The two halves of the nut hb move in a vertical direction at the* back of the piece c, and are provided with pins which fit in cccentiic slots in the citcular plate which revolves on point / Fig. 3 shows these cccentiic grooves in the plate, If the pins of the half nuts aie shifted by moving the levi ,/, the half nuts travel the double distance A B (Fig, 3), vk, : th upjnir nut up and the lower one down, the half nuts being thus jor Screw-cutting on Lathes entirely disengaged from the thiead, causing the motion imparted to the carnage by the Icadscrew to cease im- mediately. In the eailier types of consli action, with the solid nut, the carnage had to be moved by hand by means of a handle placed on a spindle in the apron, with a bevel gear on the other side of the spindle to which this handle was attached , this in its turn meshed with anothei bevel geai fixed on the hub of the nut. In this way the nut was made to i evolve ovei the leaclsciew and the cainage was moved over the bed. But it took far too long to move the cainage any distance at all ovei the bed, besides being very fatiguing work The nut, being in halves, can no longei revolve, but it can be opened. A lack is to be found on the bide of the bed in which a pinion meshes to which motion is impaited by the hand wheel h (Fig, 2), by means of which the caniage can be quickly disengaged fiom the leadsciew, and a quick and easy hand movement is secured. Other advantages besides those enumeiated here have been clenved from the split nut. One gieat difficulty, however, still remains, viz,, the diffeient weai on a ceitam length of the leadscrew If this happens to be more worn in the middle than at the ends, it is impossible to cut a true thicacl Now, in comparison to the work oidmaiily pei formed on a lathe, but little screw-cutting is done. The greater part of the time the leadscrew is thus engaged foi the feed motion of the caniage and for surfacing* For this reason, the movement imparted to the carnage for screw-cutting, has been separated from that for feed motion, A separate shaft, provided with a FIG 6 The Calculation of Change- Wheels for St.rew-cit,tting on Lathes. 7 keyway, impaits motion to the pinion which meshes with the lack (Fig 4), by means of bevel and spui-geais The sliding movement of the caniage being accomplished in this mannei, the leadscrew is only used foi screw-cutting In still latei,and principally Amencan constiuctions, the two shafts have finally been united in one, the leadscrew being now piovicled with a keyway, for sliding and sui facing the leadscicw simply acts as driving shaft, the thread of the leadscrew being only used foi sciew-cuttmg, and so the same object is attained with one shaft as is obtained in Fig 4 with two, viz., the thiead of the leaclsciew is used for sciew-cutting only FIG. 5 In Fig, 2 the gearing for the motion of the leaclsciew from the head spindle is cleaily visible. Wheel I is keyed to the head spindle ; icar wheels 2 and 3 mn loose on studs fastened to the levei 4 By means of knob 8, this lever can be laised to hole 5 or lowciecl to hole 6, If the lever is placed in position 5, wheels 3 and I become engaged, and wheel 10 on spindle 7 revolves by means of wheel 9, Wheel 2 now mns to no pin pose. If the lever is placed in position 6, wheels 2 and i become engaged, and wheel 3 is brought into play by means of wheel 2, thus causing wheel 3, as well ab wheel 9 and 8 The Calculation of Change-Wheels spindle 7 to rotate in an opposite direction In the illusli.i- tion the levei stands midway, so that wheel I engages neither of the wheels 2 or 3, consequently, although the lathe spnulk* lotates, the leadsciew is not rotating Wheels I, 2, 3 ,unl <j have the same numbei of teeth, so that the wheels on spindle 7 make precisely the same numbei of revolutions as the lathe spindle. Wheels 10, 1 1, 12 and 13 are the actual chan&e-wlu'c'ls, and can be easily mounted, dismounted, 01 changed Wheels ii and 12 rotate on a sleeve on spindle 14, and consequently make the same number of i evolutions, so that wheel \?, transmits veiy slowly to wheel 13 the motion hnpaih'd to wheel ii In the illustration the gearing between wheel 9 to the leadscrew is accomplished by 4 wheels wheels 10 and 12 being the driving wheels, ii and 13 those cluven It is evi- dent that the motion of wheel 9 on spindle 7 it. mipaited but veiy slowly to the leadsciew, in the same iatio as the product of the number of teeth on wheels 10 and 12 to the number of teeth on n and 13 Precisely the same is to be seen in Fig 4. Wheel 13 can, however, be driven by means of a wheel engaging both wheels 10 and 13, without the mteimediate wheels ii and 12, thus solving a,s an idle wheel, m which case wheel 10 is the chiving wheel and 13 the one driven The iatio between the numbei of icvolutuws ol the lathe-spindle and leadsciew is identical with the nttm between the numbei of teeth on wheels 10 and 13. Wheels n and 12 are mounted on a sleeve iimnint' mi stud 14 (See Fig. 6) This stud must be movable in accoulauco with the dimensions of the wheels, and is consequently placed in casting called the sheai 01 swingplatc at the end of the lathe This shear (Fig. 7), has two long slots, so that the stud can either be brought close to the leadscrew B, for small wheels, for Screw-tutting on Lathes. 9 or moie to the icai foi latgei wheels, at will In ordei to pei- mil of working with five or six wheels, <i second slot is to be found in the sheai This shcai turns on the le.adgc.rew B, and is held in position by means of the two bolts to be' seen in the cucular slots When the mtei mediate wheels have been accuiately set in the wheel on the leadscrew, the shcai, which was first loweied to its full extent, is laiscd till the mtci- mediate wheel engages the uppei wheel piopcily, aftci which the shear is fastened. Fig, 5 shows an Amencan type of lathe, on which it is not necessaiy to change the wheels for diffeient pitches of FIG. 7 Lhieads By means of a conc-geat to be found under the headstock and at the left-hand side of same, the latio of speed between the lathe-spindle and the leadsciew can be varied by the simple movement of a lever. The necessity of calcu- lating the change-wheels is done away with, all that is icquiicd being the placing of two levels in a ceitam position indicated in the table. The manner in which this result is attained will be {wilier described in Chapttte HI, ,* ] io The Calculation of Change- Wheels, CHAPTER II THE CALCULATION OK CIIAN(2j!.-WHI''hLS (a) System* IN the calculation of change-wheels foi sciew-tuthng on the lathe there is one difficulty, and that is, the diifeience between the English and metric system of measuicmenLs. It is not Insurmountable, but it does not icnclei the task any easiei, and has been the cause of a consideiable amount of tiouble. In the calculation of change-wheels it is <i mattei of in- difference whether a light- 01 left-handed sctew is to be cut, what form the thiead has to take, whethei the thiead is internal 01 external, 01, finally, the exact internal 01 external diameter of the thread The one essential question to be answered is How many thi cads, aic icquited foi a ceitain unit of length ? For this puipose two units exist , ist, the inch ; 2nd, the centimetre Foi both these units of length the numbei of Kvolulions of the thread are tcimed "numbei of thiedds." The length of a single thread is spoken of as " pitch." The number of thread* is thus determined dy the numbet of revolutions per ttmt of length If the pitch is indicated with the inch as the unit of length we speak of " English thiead." If the pitch is indicated with the centimetre as unit of length, it is culled a " metnc thiead " Both, however, have a system, which rs further treated of in Chapter III, but which, as such, has nothing a t all to do with the calculation of the change-wheels. , If but one of these two units, cither the inch or the centi- metie were exclusively adopted as the stunclau! unit, then the difficulty refeired to at the beginning of this chaptei would for Strew- cutting on Lathes 1 1 entirely disappeai. But the inch and the ccntimetie aie em- ployed together , and not only that, but theie is also a lack of unifoimity with legatd to the leadscrcw, one makei cutting the leadscrew according to the English, and another accord- ing to the metric system. English and American lathes usually have a leadsci ew cut accoidmg to the English system , Ficnch and Swiss makers cut it almost exclusively accoidmg to the metnc system, whilst Geiman manufacttueis employ both systems, though the pieference is given to the English. Foiu variations aie thus possible . I. A metric thread to be cut on a lathe with metiic leadscrew ; 2. An English thiead to be cut on a lathe with English leadsci ew 3. An English thread to be cut on a lathe with metnc leadsctew 4 A metric thiead to be cut on a lathe with English leadsci ew Buefly summarized To cut i. Metnc on metiic 2 English on English 3 English on metric 4, Metric on English. If one desnes, once and for all, to be able to calculate the change-wheels loi eveiy vanety of pitch, it is impel alive to know these four varieties thoioughly, as they can occur intermingled. 1st Axiom, -The number of tlircadt, ts to be determined by the pitdi of the leather ew and the uilio of the number of revo- lutions of the laths spindle to that of (he Icadscn'io This axiom holds good for all four UZ.WM. The ratio of the number of revolutions of the lathe-spindle to that of the load-screw is obtained by means of wheels (change-wheels) When the spindle of the lathe has completed one revolu- tion, then the woik on the lathe will have also completed one revolution. 1 2 The Calculation of Change- Wheels If the number of i evolutions of the lathe-spindle <uul lead- sciew rue the same, so that the leadsciew has also completed one revolution, then the camai>e has moved a distance dining this one i evolution equivalent to one thicad of the leadsctew If a tool has been placed in the toolholdei, so that it can cut the woik-piecc, then piecisely the ime pitch will have been cut on the woik-picce as that on the leadsciew. With an equal number of revolutions of the lathe-spindle and the lead- screw, the tin cad cut on the work-piece will have the same pitch as the leadsciew If the lathe-spindle has completed one full i evolution, but the leadsciew on the other hand only half a i evolution, then the carnage, and with it the tool, will have moved in a straight line ovei a length equal to half a pitch of the leadsciew It is thus only when the lathe-sptndle has made two i evolutions that the leadscrew will have completed one full i evolution, two threads aie now to be found on the work-piece over a length equal to one pitch of the leadscrew. The ratio of the numbei of i evolutions of the spindle to that of the leadscrew was 2 I , the latio of the number of thieads per unit of length of the woik-piece to that of the leadscrew was also 2 i. Hence it follows 2nd Axiom, The ratio of the number oj i evolutions of the lathe-spindle to that of the leadscrew u the same as the pro- portion of the pitch of the thread to be cut to that oj the lead- it crew Axiom 2 is also applicable to all foin cases. Foi example, the leadscrew of a lathe has a pitch of one thread to the inch It 14, lequiied to cut two threads to the inch. The piopoition of the pitch to be cut to that of the leadscrew is thus 2 i Accoiding to axiom 2 the ratio of the number of revolutions of the lathe-spindle to that of the leadscrew must also be 2 I The leadscrew has thus to complete one i evolution to two of the lathe-spindle. The leadscrew leceives its motion fiom the lathe-spindle, so that the rotation of the leadsciuw must be ictarded accoidmgly. The lotation of the lathe- spindle is transmitted to the leadsciew by wheels. The pro- for Screw-cutting on Lathes. 13 portion of the numbei of teeth on wheel 10 (see Fig. 2), to those on wheel 13 on the leadscrew must thus be in inveise piopoition to the latio between the number of revolutions of the lathe-spindle and the leadscrew, which, in the example given, must be 2 1 , the latio of the wheels 10 and 13 thus becomes I 2. If then a wheel with 50 teeth be on the sleeve of spindle 7, and one with 100 teeth on the Icadsciew, with any desued idle wheel, a sciew of 2 tin cads to the inch or }-mch pitch will be obtained on the work-piece with a Icadsciew having one-inch pitch. From this we amve at what is again applicable to all foui cases jrd Axiom The- proportion of the number of the threads to he cut to those in the leadscrew is in inverse ? atio to the pro- portion of the number of teeth on the wheel on the lathe-spindle to the number of teeth on the wheel of the lead-screw, 01 In fiactional foim Number of thicacls to be cut _ Numbei of tin cads in the leadscrew ~~ No. of teeth on the leadscrew wheel No. of teeth on the lathe-spindle wheel In this manner the calculation of the change-wheels foi scicw-cuttmg is i educed to the woikmg out of a simple fi action the number of threads to be cut being the numerator, those in the leaclsciew being the denominator, or, if it is denned to expiess the fraction in the same manner as the wheels, i e. the numbei of teeth on the lathe-spindle wheel on top as numeratoi, that of the Icadsciew underneath as denominatoi, it is just the rcvcisc. The number of threads in the leadscrew will then represent the value of the numerator, those of the thread to be cut icpiesenting the denominator As the pitch of the leadscrew on a ceitar lathe is always the same, it follows that the value of *' numeratoi is always constant, We must here call especial attention to a rmsundcrsta which so often occurs m connection with the question whether the number of threads in the leadscrew must 14 The Calculation of Change-Wheels the numeratoi or the denominate! A practical man can generally tell fairly well which wheels have to be placed on top and which underneath, but still, when the pitch of the thiead to be cut closely appioximates that of the leadsciew, mistakes can sometimes be made The screw may be denoted by the numbci of thieads pet unit of length, m which case the numbei of thieads in the leadscrew is the numerator of the fi action The screw may also be denoted by the length of one pitch of the screw , m this case the length of pitch of the scicw to be cut will be the numeiatoi, the length of pitch of the lead- screw being the denominator of the fiaction, the numeiatoi of which will indicate the numbei of teeth on the lathe-spindle wheel, the denominator indicating the number of teeth of the wheel on the leadsci ew Should the numbei of thieads of the sciew to be cut be a multiple of those m the leadscrew, one is naturally inclined to express it in number of thieads per unit , foi example, 4 threads per inch to be cut on a lathe with a leadscrew of i thread per inch , should it not be a multiple, as for example, each thread having a length of 7 mm., one is then inclined to denote it by the pitch. If, m both instances, the number of threads in the leadsciew be r pci inch, the fiaction in the first instance will be Number of thieads in the leadscrew _ , _ chiving wheel Number of threads to be cut wheel to be d liven In the second instance, in which the pitch of the screw to be cut must be 7 mm., the number of the threads to be cut per unit is itself a fraction, viz - 5 4 , the fraction thus being i 7 . 7 being the length in mm. of the pitch of the fo _4 ^3 4 7 ~ screw to be cut, 25 4 the length in mm. of the pitch of the lead-screw, so that, in this case, the length of pitch of the screw to be cut can at once be placed m the numeiator foi the driving wheel, the length of pitch of the leadscrew being for Screw-cvitttng on Lathes 15 placed in the denommatoi foi the wheel to be duven In actual calculation the fotegomg examples must be carefully distinguished one from the othei (b) What Change-wheeh aic to be found on a Lathe This question piescnts itself each time change- wheels have to be calculated, because the fraction which is formed by the thiead to be cut and the leadsciew, must be changed into one foimed from the wheels to be found on the lathe These wheels should have such a number of teeth as will, within ceitam limits, include the indivisible factors, viz 2, 3, 5, 7, II, 13, 17, 19, 23, etc Some makeis supply these wheels in a piogiession of 5, othei s with another progression. The following set of change- wheels is, or should be piovided with eveiy lathe, vu 15 = 3x5 60 = 2x2x3x5 20 = 2x2x5 65 = 5x13 25 - 5 x 5 75 = 3 x 5 x 5 30 = 2 x 3 x 5 85 = 5 x 17 35 = 5 x 7 95 = 5 x 19 40 = 2x2x2x5 100 = 2x2x5x5 45 = 3 X 3 X 5 105 = 3 x 5 X 7 50 = 2x5x5 115 = 5x23 55 = 5 x ii _ 125 = 5 x 5 x 5 01 If) = 2X2X2X2 42 = 2X3X7 18 = 2 x 3 x 3 44 SB 4 X ii 20 = 2x2x5 56 = 2x2x2x7 21 = 3x7 60 =2x2x3x5 22 = 2XH 66 = 2X3X11 26 = 2x13 78 = 2x3x13 28 = 2X2X7 88 = 2X2X2X11 34 = 2x17 96 = 2x2x2x2x2x3 38 = 2x19 1 08 = 2x2x3x3x3 One of the two foiegoing sets is gencially provided with 1 6 The Calculation of Change-Wheels the lathe English lathes usually have a set of 22 wheels some of which have the same number of teeth It will be clear from what has been said, thus far, that the easiest thiead to be cut on a lathe, i e the thread causing the least tiouble m the calculation of the change-wheels, is that having the same system as the leadsciew This will be the case with the ist and 2nd cases refeired to on page II (c) The Cutting of Metric Threads on a Lathe with Metric Leadscrew Take the case of a lathe with a leadscrew having I cm. (10 mm ) pitch. It is requiied to cut 4 thieads per cm. No of teeth on driving wheel _ No of thieads in the leadsuew No~o7teeth"oia wheel to be driven ~~ No, of thieads to be cut = 1 = _*L = gear-wheel 10 1 Sce F]g 2 4 100 = geai -wheel ou lead-screw J It is required to cut 7 thieads per cm. No of thieads in the leadscrew _ I _ 15 = driving wheel. No of threads to be cut 7 105 = wheel to be duven To cut i^ thread per cm. No of threads m the leadscrew ,_ j_ _ 5 or 6 = cluvmg wheel. ~No of threads to be cut 15 75 go = wheel to be duven. To cut 3 threads per cm No of threads in the_leadsciew I _ 30 = driving wheel No of threads to be cut 3 90 = wheel to be driven To cut 5 threads pei cm. No of threads in_the_leadscrew I 25 = driving wheel. No of thieads to be cut 5 125 = wheel to be duven. In the last example it is also possible to say, a pitch of 2 mm , in which case the fraction will be Pitch in mm to be cut _ 2 _ ^25 = duving wheel Pitch in mm of leadscrew lo 125 = wheel to be driven. In both cases the result will natmally be the same. Jor Scrcw-cnttuio on Lathes. 17 Pitch in mm to lie cut _ 7 _ 70 = duving wheel I'Ucli in mm of leadseiew 10 100 = whcc>l to he duven Pilch in mm to ho tut _ 5 5 _ S5 = dnvm^ wheel Pilch in mm ol k'idseiew 10 100 ~- wluel to In duvcti To cut a pitch of 7 mm Pitch in mm to be cut Pitch in mm of leadseiew To cut a pitch of 5^ mm Pilch in mm to ho tut Pilch in mm ol k'idseiew To cut 7 threads pei 22 mm 22 Denoted in pitch = a pitch of " mm Pitch m mm to he cut _ "/- _ 22 ~ dnvinu; wheel Pitch m mm ol leadsuew TO 7 " wheel to lie duven ,L i zot Fu>. 8 Fiu. 9. So fai it has always been possible to work with a single tiam of wheels with any desued idle wheel. Fig, 8 shows a single train. 1 8 The Calculation of Change-Wheels In the set of wheels to be found on the Lithe, wheels with cithei 22 or 70 teeth, as piesumed weie employed foi the pieccding examples, weie not included A compound tiain is now used. 22__2XH _ 20x55 dnvtntf \\liccls 70 ""7x10 ~ 35X HID wheels lo lie dnvcn Fig 9 shows this comj)ound tiain a and /; are the drivers, c and //those d) iwu The fixing up of the wheels will thus be 55 x 20 100 x 35" The wheels m the numeiatot, as well as those in the denominate!, can be interchanged , a may thus be put in place of b, 01 c in pLicc of </, or both may be c lumped , btit interchanging of a clnvei wilh one to be diiven may nevoi take place, as this would altei the value of the fraction and an entirely different thiead would be obtained. It is always advisable to tiy to get the smallest of the drivers on the lathe-spmdle, and the laigcst to be driven on the leadscrew, in oidci to obtain as lational a geating as possible. To cut 1 1 threads pei 14 mm I he pitch is thus 14/11 mm, Pitch on leadsciew ro mm. Solution 2X / ro n x 10 j i x 10 55 x 100 To cut 3^ thread pci 40 mm. The pitch is thus 40/3 * 5 mm. Solution 40/3" 5 = 40 ^ 4 X 10 ^ sox IOO ( 10 3^x10 SX; 5^X35 To cut /] tin cads on 15 mm The pitch is thus 15/4 nun. Solution IS ^3X5 ^30x50 f 10 4 x 10 4x10 40 x ioo Should the lathe have another pitch than 1 cm., this will only necessitate a change m the constant of the leadsercw In the fi action. for Screiv-cutting on Lathes. 19 The following aie a few examples with solutions, dealing with diffeicnt leadsctcws To cut 9 thieads pei 16 mm , leadsciew 2 threads per i cm The pitch of the thiead to be cut in 16/9 mm The pitch of the leadsciew is 5 mm. c , , 1 6/9 2x8 20 X 40 . . , , Solution ' - = ^ = in case these wheels 5 9X5 45X50 , n 20x80 are too small 45 X 100 To cut a pitch of 3 mm. Pitch of leadsciew being- 7 5 mm. 2 -2Q Solution 7 f 5 = ^ To cut 8 threads pei 13 mm Pitch of Icadscrcw, 7" 5 mm Solution 13/5 '3 =2x6 5^20x65 75 8x75 8x7-5 80 x 75 In both the foicgomg examples, a wheel with 75 teeth appeals among the wheels cliivcn, but is not included in the specification given on page 15. With a leadsciew having a pitch of 7' 5 mm. a wheel with 75 teeth will icpcatcdly occui , in such a case the manufactiuci will be certain to supply a wheel with 75 teeth. To cut a pitch ot 20 mm Lead screw pitch 25 mm c . 20 100 Solution 25 = 125' To cut 3 threads per 20 mm. Leadsciew pitch 25 mm. t , . , 2O/3 2O 2 X 10 40 X 50 solution f.^i ~ ~ solution. 25 - 3 x 35 - 3 x as - fo x I2S - To cut a pitch of 37-5 mm. Leadsciew pitch 25 mm. Solution. 37-5^15X25 30x100 on. 25 10 X 25 40 X 50 To cut a pitch of 76 mm. Leadscrew pitch 25 mm. 76 4 x 19 40 x 95 80 x 95 Solution _ 35 2'SXIO 25X50 25X100 C 2 2O The Calculation of Change- Whech (d] The Cutting of English Threads on a Lat/ii unfit English Lcadscrctv In principle, this second case lescmblcs the fiist The system of the leadsciew and the Ihicad to be cut is the same. Most lathes have a leadsciew with ^ in pitch, thus 2 threads pei inch Heavy lathes have a leadsciew with i in pitch, the smallei sizes in, or 4 thicacls per Inch, whilst in exceptional cases 2\ thieads pei inch aie to be found. Given a ceitain pitch, the ft action can then be clctci mined without any difficulty Should the screw be denoted in a ceitain numbei of thieads per inch, the number of tin cads pei inch of the lead- sciew is placed in the numerator, the nunibci of thieads pei inch to be cut in the dcnorninatoi Should the sciew be denoted in the length of the pitch, then the length m inches of the pitch to be cut is placed in the numeiator, the length m inches of the pitch of the leadsciew being placed in the denominator. In practice the majority of threads aie cut accoi cling to the Whitwoith system (see page 57), for which reason we shall fust of all give a number of problems with solutions fot this thread. To cut I in Whitworth thread Leadsciew 2 thieads per inch J in Whitwoith thread = 16 thieadb per inch. Solution No of threads in_le_adscrew P C1 " inch No of thieads to be cut pei inch = 2 = _?! *_L _ _ 2 1 * S 16 16 x 12-5 ~~ 80 x 125' To cut ji in Whitwoith thicad Leadsciew 2 thieads per inch i\ m Whitworth thread = 7 threads pei inch. Solution 2 = - 2 -*_L 2 .X. 40 7 3*5X2 35 x 80 for Screw-t,nttmg on Lather. 21 To cut 2 in Whitwoith tin cad Leadsaew 3 threads pei inch 2 in Whitwoith tin cad = ^ thiead pei inch c . 4 2 2 X IO 40 X 50 Solution: = = J 45 5X9 45 X 100 To cut 3 in Whitwoith thiead Leadscu-w 2 tin cads pet inch. 3 in Whitwoith thiead = 3,\ thiead pel inch c , 2 40 bolution : = - . 3'5 7 To cut ij in Whitwoith thiead Leadsaew ,j tlueads per inch I \ in, Whitwoith thiead = 7 tin cads pei inch. Solution = 7 70 To cut i^j in. i>as tin cad. Loadsuew 2', thiead pel inch- 1 1 in jjas thiead = II tin cads per inch ,, , t 2\ 2S 20 X JO Solution - = = * . ir no 5 5 x 80 To cut 2]- thiead pei inch, Loadsctcw 2 tin cads pel inch. c , . 2 2X4 2 X 'I 20 X ()0 Solution - r =- = ' = 2 ! 9 3X3 3^ X 15 To cut I thiead pei inch (not it 1 indi pih //), Leatlsciew 3 tin cads per inch c , ., 2 2 X H 80 .10 X tOO Solution w sss s=s ss. H 7 35 25 x 70 To cut 2\ thiead pel inch. Londscicw 2^ thiead per inch. Solution; 3 i aa ]ro sa| 50 i 2% n 55 In the following examples, the length of pitch is given. The pitch of the leadsciew will consequently appeal in the denominator. 22 The Calculation oj Change- IV liceL To cut a j in. pitch Leadscicw 2 thicads pei inch =t ^ in. pitch. Solution I^H^fo' 1o cut a | in. pitch. Lcadbcrevv 2^ tin cads pel inch a= r in. pitch 2\ c . . }-?, }f. x 2-V it; x i\ 75 x icx) Solution -- = ' (> = > ~ = x J ., . i i 16 x i 40 x 80 2& To cut 19 threads on 11-5 in Leadsciew 2J, tlneads per inch. The pitch to be cut = in. The leadsciew ly pitch ~ in ^^ ii'5 Solution -I2_= !I 5x2.5^115x1125. _ 19 95 x 100 2^5 To cut a pitch of 4 % in. Leadsciew I pitch per inch. Solution: l5 f = 3 X 13 = 6 5 X 75. i 8 2x4 25 x 40 To cut a i J in pitch Leadsciew 2 thicads pci inch. Solution il = I 3X2 == 2x I 3 _65 t i- 32 2 X 1C 80 To cut 9 threads per 5^ m. Lcadsciew 2\ tin cad per h The pitch leadscrew = -v in. inch The pitch to be cut = ^ in The pitch of the d La Solution JL _ 5if! X 2& ^ 95 x_2/5 125 v 95 _i 9 9 X 16 80 x 90 " 21 In the foregoing examples practically every case which Is likely to occur, has been treated. for Screw-cutting on Lathes. 23 (a) To cut English Threads on a Lathe with Metric Leaded evv In the first and second cases considered, the system of the thread to be cut and that of the leadsciew weie identical, viz., in the fust case accoidmg to metnc measurement, in the second, according to the English measuiement In the thud case, howevei, the system of the thread to be cut and that of the leadsciew aie dissnnilai The leadscrcw is divided per cm = 10 mm., or some pait 01 multiple theieof, the screw to be cut being divided pei inch = 25 4 mm., 01 some part 01 multiple theieof In the third case to be considered, this numbei 25-4 will consequently appeal icgulaily eithei in tne numciatoi 01 the dcnommatoi, and will mvauably produce a fi action which, with one exception, cannot be lesolved into whole numbers An equivalent must theiefore be found, by means of which it will be possible to foim a divisible numbei from the nume- rator and denominator of the fraction This equivalent is to be found as follows 6^ in. = 16-509675 cm., taking for gi anted that 6\ in. = 16-5 cm., there is then a disci epancy of 0-09675 mm. per 165 mm. of length, 01 rathei less than 0-06 pei cent., a diffeience of piactically no mipoi Lance whatcvei. If the number of tin cads to be cut be expressed in a certain number pei 6 5 in , and the number of threads of the leadsciew be also expiessed in a certain numbei per 6-5 in. or 10 '5 cm., the result will be an equivalent which can be made use of. As reference is heie made to a ceitain numbei of thicads per unit of length, in this case, 6-5 in. 01 16-5 cm., the numbei s of thicads of the" leadscrcw will appeal m the numeratoi, the number of thieads to be cut in the denominator. The following comparison can thus be formulated No of threads in leadsciew per 16" 5 cm, ^ chiveis No. of threads to be cut per 6-5 in, wheels to be driven 24 The Calculation oj Chaugc-lVhccL As the numbei of thicads in the leadscicw lenuuns in- variable for the same lathe, the numcuitoi is consequently a constant factor foi a ceitain lathe, Should the leadsciew have a I cm pitch, the hvulsciew will then have 16-5 thieads pci 16 5 cm , and the constant factor of the numciatoi will be 16*5, whilst, at the same time, 6* 5 is to be found as a constant factoi in the denominatoi, and must constantly be multiplied by the number which expi esses the number of threads to be cut pei inch I( both these constant factois be multiplied by 10, the numbei 105 will always appeal in the numciatoi and the numbei 05 m the denominator, in this way constant factoi of numerator 165 denominator No of thieads to be cut pei in X 65 or .. ll x J 5 threads pei m X 65 The equivalent is now complete , by icplncnu* threads pet znckm the denominatoi by the actual numbei, a fi action is obtained which will permit of the calculation of the wheels, In the examples which follow, eveiy possible vaiiation has been carefully worked out, fiom the simplest to the most intiicate To cut 6 threads pei inch. Leadsciew 10 mm pitch, Numeiatoi = _ 0x15 11x15 Denominator = No of thieads pei inch x 65 ~" 0x65 = 55X75 = 50X55 150x65 65x100* To cut 4 threads per inch. Leadsciew 10 mm, pitch. Solution IIXI 5 = 11x75^55x75 4x65 20x65 65x100' To cut 2 7 | tin cads pei inch Leadsciew 10 mm, pitch. Solution IIX 15 = 4Xnxi5 = 2X2X 3x5x11 4X65 9x65 3X3X5X13 -, I2X 55 ^ 55x6o 45X13 45x65* for Screw-cutting on Lat/tes. 25 To cut 5\ tin cads per inch Lcaclsciew 10 mm pitch. c 1 4 11x15 1 1 X 30 30 Solution ; r = ~. = , 5 5x65 11x65 65 To cut I in Whitvvoith-thiead = 8 thieadb per inch Lead- bcicw 5 mm pitch. In this case the Icadsciew has 2 tin cads per cm Conse- quently for this p<irticulai lathe, the numeiatoi ib 2 x 165 = 330 01 II X30 Solution 11X30^60X55 8 X 65 65 X So To cut } in yat, thieatl = I4thiea ( ds per inch Lcadscrcw 5 mm. pitch 014. ii xso 30 X 55 Solution ~ = , J . 14x65 65x70 To cut -]_ in. Whitwoith -thi cad = 20 tin cads per inch. Leadscievv 5 mm. pitch, c 1 4* II X3O 30X55 Solution J = ; " . 20 x 15 65 X 100 To cut I in. gas thiead = II thieadspei inch Leadsciew (> mm. pitch No. of threads in leadscrew per cm., ^P. Solution " ' SAA " t3 = " ; i5 =-7 V 5 =r 5 - 11x05 65 6x65 05 To cut 36 tin cuds pei inch. Leadscrew 4 mm pitch. No. of threads in the leadsciew per cm , l t 01 2*5. Solutmn. -5xnxi5 1 .nx-5_ 35X55. 36X05 I ' X' 3 ' T X 120 To cut I thread per inch Leadscrew 10 mm. pitch. IT x 120 _ 55><i7o (""* -I . JU J /\ IT ^' "n * * <\ * boluuon. == 7x( . 5 _ 7xfl5 - J5X65 26 The Calculation of Change-Wheels To cut a in pitch Leadsciew 10 mm pitch. No. of threads per inch = = . 77 8 / Solution * = ' -* = ^ 8 - 8x65 65x80 7 To cut 3 thieads pel 2 in. Leadscrew 6 mm. pitch No of threads pei inch i| No. oi thieads in the leadsciew per cm \ c 1 4. V x ii x 15 10 x ii x 5 55 x 100 Solution . (> , . -*= 3 = i- 3 J x 65 3 x 65 30 x 65 To cut 36 threads per 7 in. Leadscrew, 7 mm pitch. No. of thieads per inch ty No of threads in the lead.screw per cm \ Q . Solution- Vxiixis^ioxii Xi S a 7 b X 65 36 X 65 _ 5^x n _ 50 x 55 6 x 13 60 x 65" To cut 9-5 tin edd per 8 inch. Leadscrew, 10 mm. pitch. No of thieads pei inch, ^ 5, - TI X 15 ^ 8 X ii x 15 _ no X 120 Solution- To cut 25 thieads pei 3^ in. Leadscrew, 5 mm. pitch. No. of threads per inch, |i = Io . No. of thieadb in the leadscrew 3i 15 per cm, = 2 Solution 2 x u x I S = ^ X ii X 15 X 15 Vy x 65 ioo x 05 = 5S x (JO . 65 x 100* for Screw-tutting on Lathes. 27 To cut a 2J> in pilch. Leadsciew, 10 mm. pitch No. of I 2 tin eads pel inch, = . , A . 11x15 5x11x15 1 10 x 75 Solution -, , w = , = / ^ X 05 2 x 65 20 x 65 To cut 2 tin cads pet 6,1 in. Leadhtiew, 25 mm pitch No, of thieads pei inch, ~ = ^ No, of t hi eads in the lead- I 2 sciew pei cm,, ^f~ * " i 5 t , . ^xiixi 1 ; 2xi>xnxi5 5sx 60 Solution ' , . J = - i x ot; 1 X 5 x 05 k |u x 2t; (/) 77^' Cutting (>J Mt'tm Tluead^ on <i Lathe witk To some extent the fouith case lesembles the thud The piopoition 10:25*4 also holds good, though with an opposite meaning Use is also made in this instance of the fact that 6" 5 in. is equivalent to 16*5 cin Suppose, for example, that the leadscrew has a t inch pitch and 10 tin cads per cm. have to be cut, i.e. a I mm. pitch, then, when the leadscrew lias completed 6-5 revolu- tions, the lathe spindle should have made 165 i evolutions, which can be formulated No. of threads in the leadscrew pci 6-5 in, _ 6* 5 No, of threads to be cut pet 165 mm. ~~ 105* The numerator of the fraction will thus, for a yivcn lathe, always be equivalent to the number of threads per Inch in the leadscrew X the factor 6*5; the denominatoi being equivalent to a fraction, the numeratoi of which is the factor 165, and the length in mm. of the thread to be cut, the denominator. 28 The Calculation of Change-Wheel \ Foi example, a 2 mm. pilch is to be cut on a lath having a leadscrew of 2 threads pei inch, then the numeiator will be 2x6 5 = 13 and the denominator will be 2 Foi this paiticulai lathe the numeiator will always be 13 The fust lesolvent of the fraction is a whole mmibc obtained fiom the clenominatoi by placing the denominate) of the fi action, winch is the denominator of the compouru 2X1^? fraction in the numeiator, thus r No useful puipose, howevei, is effected by this alteratioi eveiy time The pitch of the thread to be cut is according!) placed directly in the numeratoi, the fraction then benu definitely foimulated as follows Numeratoi = Pitch m mm, of thiead to be cut x No. of thieads in the leadscrew pei inch X 6 5 Denominator = 165 Attention must here be directed to the fact that whenevci the length of the thread to be cut is a fraction, it must nevei be lesolvcd into a decimal, but must always be placed in the numerator as a vulgai fraction, so that compound fraction 1 - maybe icsolvable from numeiatoi and denominator by multi- plication of both. The following examples, from the simplest to the most complicated, will make cleai what has been stated above . To cut a bciew of 5 thieads pei cm Leadscrew 2 threads pei inch To be cut a 2 mm pitch. Solution. 2X?X6- S= 2x13 = 20x65 , 165 ii x 15 75 x no To cut a 3 5 mm. pitch. Leadscrew 2 threads pei inch Solution. 3'$xr 3 = 35X6 5> 11x15 75x110 for Screw-cutting on Lafhc\. 29 To cut a sciew of 3 tin cads pet cm, 'Leadseiow 2 thicads pci inch , To be cut a ^ mm pitch Solution ^ X1 3 = 10x13 = 10x1^20x05 11x15 3x11x15 11x15 ,15x110 To cut a screw of 8 thicads per I i mm Lc.ulsetcw n threads pen inch To be cut a y mm pitch c , . V x 13 13 20 x 05 Solution - h = = J . 11X15 8x15 J GO X \ "M To cut a sciew of 5 thieads pei 18 mm Le.ulsdew > thicads pci inch c i 4 M^xis 13x18 6x13 s <! xfK solution __ - - = r- ~~ * " 11x15 5x11x15 nx^ 55x115 To cut ,i sciew of 4 thicads pci 7 mm Lisulsucw ?\ thicads pci inch Solution 1X2^X6^ 7x13 35X^5 . 11X15 4X2X6X11 I 10 XI 30 To cut a 7^ mm pitch Lcadsciew 2\ tin ends p<jt inch. Solution , 7 ^ x 2 l*fia = 5 X J 3 ^ 5<> X 65 11x15 tixS 5 5 x tSo To cut <i io| mm pitch. Lcadsciew i thuNid pci inch, Solution 10 * x(5 i 2IXI 3 ^ 7XI3^3SX65 LIX15 4XI1XI5 11X30 55x100' To cut a 42 mm. pitch Leadncrew i inch pilch. Solution 42X6-5 = 42x13 ^ 7X rj ^ 70x65 11x15 2x11x15 sx n ~ 50x55' To cut a sciew of 13 thieads pet 5 mm, Lcadscnnv /j thicads pei inch. Solution, x/ix 2XS ^ 20x25 11x15 7-SX22 75x110' Illlllllllllllllllllll 30 The Calculation of Change- 1 7 '//<'<?/ r ^ (g) 7/# Wheel with i 7 /re//' In addition to the equivalent 6* 5 in. i<VS < m,, nhirh has been employed in the thud <uid fourth cases, theie is still another way of cutting English thu-.ul on a lathe with metnc leaclsciew, 01 vice vet&d, winch is, by nicking use of a wheel with 1 27 teeth The pioportion between the cm. and the inch of 10 2}'4 can be resolved into one of 50 : 127 127 is not divisible fuilhei, <ind so, if a wheel with 127 teeth be employed, this i.ictoi can be- placed eilhej in the numcratoi 01 the denominate! . The thncl and fouith cases will then leseinble the fiint, seeing that it is now possible to expiess the Knghsh tlmsul in mm, whcthei it be the thieads in the leaclscivw r the thieads in the sciew to be cut The ft action will thus be Numeratoi = Pitch to be cut in mm Denominator = fitch of leadsciew in mm Numerator = No of threads in lead screw pet inch Denommatoi = No. of threads to be cut pei cm X 2*5-1 The following examples will cleat ly indicate* what Is meant To cut a 2 mm. pitch. Leadsciew 2 thieads per inch, Leadscrew pitch 12*7 mm. Numeiator = 2 _ 20 Denommatoi = 12 7 127 The foregoing example, when wotked out as per the UiHt comparison, will yield the same result, seeing* that : - 2 mm. = 5 threads pci cm. Numeiator = 2 __ 3 __ 20 Denommatoi = 5x2' 54 ~* 12-7 "~ 127 for Screw-cutting on Lathes. 31 To cut 3 threads per cm Leadscrew 2 threads per inch. Solution - -- -= ~ 2 ___ = 4X50 = _40>5o_ 3x254 6x1-27 6x127 60x127 01, accoidmg to first comparison, i pitch = 10 mm. Numerator = ip j; O _ 40x50 Denommatoi = 12 7 ~$xi2 7 Y~~6ox 127' To cut 7 thieads pei 44 mm. Leadscrew 2 threads per inch 44 Solution -L_ = _J4_ = 40 x 55 12 7 7x 12-7 35 x 127 To cut a 9 mm. pitch Leadscrew 2^ threads pet inch Solution -2-.= 9X25 = 45Xi2 S . 25 '4 254 50x127 To cut 28 thieads pci 45 mm. Leadscrew 4 thieads per inch. Solution. JL= 45X4^45X50 25-4 28x25-4 70x127 4 To cut i m Whitworth-thrcad = 8 threads per inch Lead- screw 10 mm Pitch to be cut = ^4. mm. 8 25-4 Solution: _i_ = 25 '4_ = 20x127, 10 8xio 10x100 When cutting metric thread on a lathe with English lead- screw, the wheel with 1 27 teeth is always to be found amongst the wheels driven, whilst, when cutting English thiead on a lathe with metric leadscrcw, it is found among the drivers. To cut 3 in Whitworth-thread = 3^ threads per inch. Leadscrew 10 mm. pitch. c , . 25-4 20x127 Solution ; - -- J ~ = 3*5Xio 35x100 32 TJic Cal( illation of Chtuigc-U7iec/\ Yci cut 4 in, <^as thicnd = 11 tlueads pei inch. Leadsnew 14 > nun. pitch e i , 25 4 20 x 1:7 Solution J h = ' I I X 1C) IOC) X I IO Io cut 3 tlueads pei 8', ui =- nu h pil<h Lcads(io\v o ', |l > nun. pitch 25-4 3 r o 3x10 2 5 x ' 10 To cut <) tlueads pci 1 1 in Le.ulsuew 25 nun pitch, 25 A 1 1 Solution 9__ ^0x25-4 = 45 xi -7 25 11X25 55x125' To cut 7 tlueads pei 3 in. Leadscrew 7 nun pitch. "^^^C'/l ^ O V I ^ 7 Solution ^ "- 1 ^ = " 7X7 35X70 To cut 24 thieads pci c) in. Leadscrew 5 mm. pitch c i < 9x215-^ /is x 127 Solution = 1J , 7 2/1 X 5 50 X 60 (//) Melltod for Calculating Approximate F Bcfoi e conimcncinjy wit IT, the actual calculation, the question was pi opoundcd nuclei heading (//) on p<i t i>e 15 . " What chanp.e- whools ue to be found on <i lathe?" This was indeed im~ ju'tMtive, as the change-wheels actually pie.sent on the lathe have invuiubly to be taken into account, first of all because the fraction must be lesolved into munbcis coucspondinjjf to the change-wheels, and then, because the same faetois which *u to make up the fraction must also be found in the change- wheels. Should the fraction contain a fac toi not to be met with In the change-wheels, then, aceoiding to the methods now in vogue, a suitable set of wheels could not be iound, for Sirew-cuttnio on Lathc\, 33 consequently, the thicad in question could not be cut without obtaining one 01 mote \\hecls making up the lequisile faclois, which, of coin so, would not be possible, as a certain thiead is ijeneially tequired to be cut without notice, and theie is, theiefore, no chance of eithei making 01* obtaining suitable wheels, Will siuh eases often ocem ? Not as a nile The examples aheady ^iven cleaily show that even HI the case of thieads which vuiy veiy consideiably, the wheels necessaty foi cutting ,i line thiead can be found In the set of change-wheels, j^iven on page 15, the following fact 01 s weie found 2,3, 5,7, n, 13, 17, 19, 23; the factoi 33 was not met with in the second set, whilst on many lathes UK far to is 17, i<), and 23 aie absent If jetton ap/hw in //if fnution composed of the thread lo fa' cut and the /twhcniv, wJiii/i cannot />e found ut the diau^e- w/ft't'/s, then sitth n thififd tannot be ait accurately, If it is absolutely necessaiy to cut such a thiead, a ft action must be sought lot which appioaches the coirect fraction as neaily as possible. Lack of knowledge of the conect method of finding out a I rarl inn apptoximalimj the tine fi action as closely as possible*, loo often tesults in the calculation be-in}; skipped ovei, and a li.ution btnn t t chosen which actually ^ives a thiead diffeiint; rtnsiderably fioin the one tequired. In addition, the fact ts loo often lost si^ht of that an approximate fraction will still result in an uuseiviceahk thiead. Support", for example, a fi action is found which yields a tturad tlifiVrhtjLf only 0*05 mm, fiom the thread of the nut to lit which the thread has to be cut. At first si^hl the differ cmv appears trifling, but the en or which has been made is ieally vwy tfj't'trt, so ^reat, indeed, that the thtead obtained is wholly #,w'/d'.vx It must of course not be foi|,;otten that each thread inct cases the error, which at the end of 20 tin ends will result in a difference of 30 X 0-05 inm, = i mm Suppose, fwther, that a threatl h.ts to be cut of 23 threads per inch, 1 2 OQ the pitch being * 1 4 " tnm With a difference of 1 h 25 ',1 25^ I) 34 * The Calculation of Change-Wheels 0-05 mm pet thread, the diffeience at the end of 10 thicads will be equivalent to one-half of the thiead, whilst at the end of 23 thieads, the diffeience will amount to the en the thread The foiegoing example clearly demonstrates that only fractions diffenng by some thousandths of a millimctie, or some ten thousandths of an inch, can be employed How can such an appioximatc fraction be arrived at? 1 Regular piactice often enables one to find a fi action which approaches veiy closely, without the assistance of any method In one of his note- books the wntei found a fraction which had been discovered, apait fiom any method, foi the cutting of a 3 7 mm. thiead on a lathe with a leaclsciew having a pitch of 10 mm For this thread theie were no change-wheels, for a wheel in which the factot 37 appears, which is indivisible, is not to be found among an ordmaiy set of change-wheels Foi this reason, according to the notes in question, the 77 fraction -~ was chosen, for which change-wheels could DC f , found, since 7x11 35 X 55 * -_- ---- -= - - 208 13 X 16 65 x 80 q 7 y? Seeing that the difference between -~~ and ~ is simplv 10 208 L J the difference between 3-7 and 3*701 = o ooi mm., so that aftci 10 thieads the diffeience is still only croi mm., which may be considered neai enough for all practical pin poses Such gi oping about in the dark, however, is not at all methodical, can take a veiy long time, and, finally, may not lead to any actual icsult. The compound fraction, however, supplies us with a leacly means of discovering a fi action which appioximatcs suffi- ciently to pei nut the obtaining of what is piactically an accurate thiead. Suppose the fi action to consist of two numbcis, the numciator and denominator of which arc both positive for Strcw-tittting on Lat/ies 35 Let these numbois be lepiesented hy A and B, and A > B This can then be tepicsentcd ^ =s rti 4- 'l /', < B 01 B > ;-! Taking the icveisc of the last-named fi action, the i eduction ran then be huthei continued, K , ''' ^ = (h 4- , /j < ?i 01 i \> >'> Continuing fuithei which can be continued ad infimtum, and can thus be expiessed /-, " '-.' in which ),,<>,< i <>' ''// i > >w The (juotionts /?,, ^ 3 , ^. </*, ire termed indicators, By substitution can be obtained A r a i + ', etc., etc. i i i x ' - 1 1) 2 36 The Calculation of Changc-Wheeh If ._ o, then the numbci of tenus is finite, in which r*-i A case the fiaction is detcimmable, in that it can finally be divided without leaving a icmaindci. A If the pioportion be mdctciminablc, and cannot con- sequently be cxpiessecl by a fiaction with exactness, then theic will be no end to the divisions, in which case the number of teims of the compound fraction will be infinite Every indeterminable numbci may be ief.aided as the Until of an indefinite, non-iecumni> fiaction The /////// of a lepcatmg decimal fiaction is a determinable ptopoition, ei>, the limit of 0-3 is J. To apply the foicgoing to a definite fiaction. (i) Given A > B, for example To expiess the fiaction as a compound Itnclinn 6961 6961 4- i + i 3 + 93 1 -f i A + " 2'3 -h 1 6+ 7 The indicators aic thus I, 2, 3, /j, 5, 6, 7. for Screw-cutting on Lathes. 37 Consequently ^ 7 as a compound ft action =. , 6+ * 7 2+ ---- * + j \ 4.4- (2) Given A < B, foi example. To express -^ as a compound ft action. H3 _. _1_ _. i 355 355 , _i6 113 3 ~ l " 113 + i 7+fg A If ..- <; I, the first mdicatoi can then be expiessed by o, in which case the mdicatoi s will be o, , } and T \ r , thus 113 i , ,. - = o -f- as compound fraction 355 * (3) Express the compound fraction 4 as an ordinary fraction. 3 2 i 4 A = 4 + X == 4lt 3 + 2 + \ -3* S 38 The Calculation of C//ti/i^i'-ll iin A (4) Expiess the compound haetion \ as an oidnuiy fi action ' A B -f 3 + = A + 1 T ~ (i A 1 03 1? is thus = B 215 The genual fonnula can now be expicsscd by putting Icttcisin place of the figures given in the, foic;,Mmi;,> examples. Given the compound daction a, determine the ouliiuuy fraction b A given that a i b = 2 c = 3 </ = 4 a + c<t d a />( d \ a b \ </ (/ i t d but I b I d bed -I- /; -f d cd -I T l Cd H- 1 </ " ;/ then lu d s= 34 = 2 ^/= 4 c d &= 12 bed =, 24 b 2 - the 1 nuuu'iator. 30 = the denominator. so that in this ease the value of the fi action A ' B Jor Sit'cw-in'titig OH Lathes 39 1^)1 any given value of a, l>, t, and d, the fi action can be imim-diaU'ly detci mined from the ft action a hi, d + a b -f a d 4- c, d + I b t, d -f- b + </ To take the level se Given the otdinary fraction (a b -f i ) 6 4- d />i+ i dcU'imnu' the compound d.iction. (a b + I ) ( 4- _. rf d <? + t -f a /^ + 1 /><+! = a -\ = indiCtilois. *+' c 'I'he indicators ate thus a, b, and c (liven that in the foiegomg fi action the indicatois have the following value a ~ 2, b 3, c = 7. Then revcisnig the oidei of things in the foiegomg example (a b + t) c + a _ (2 X 3 -|- i) 4 + 2 _ 28 4- 2 = 30 bt -\- i 3x4+1 12-fi 13' ^O The inclicutoih foi the h action ^ aie thus 2, 3, and 4, The foregoing consequently pioves (1) That uvcty detcrtiiinablc fraction may be expressed as ,i finite compound fraction. (2) That every finite compound fi action maybe expiessed us a (Interminable fi action. Compound factions may be divided into: (a) Symmetric. SX 4<D The Cah ulation of Change- \l r hceh If teims and compound fi action be cxpiesscd as B = (^" ai a "~ indicators then A is tcnned a symnictuc compound fi action because the in dicatois end in the same oidei of sequence as they began , and is teimcd a penodic compound fi action, bee , uuse the uulu atots a,,, a it rtj occui peiiodiudly In both cases the ntunbei of terms is infinite 77/# Findmg-ont of ApptMimating Whcncvei the factois of a fi actual, act 01 ding to winch a thread is leqimed to he cut, aie not rcpicsented by the change-wheels belonging to the lathe, it is impossible, us bus ah each/ been dcmonstiated above, to cut a theoretically acctnate tluead, but an attempt can be made* to discover a ft action, the value of which appioaches that of the teal ft act ion so closely that the two may be icgardecl as piaetiealty identical, Such an approximating ft action can bu found by resolving the fraction into a compound fraction, and toinunatlnp, this at the second, thud, fouith, fifth, etc., indicator Foi example A . i = ^ 4. jor Scrctv-uiftmg on, Lather Koi the fust quotient substitute then the second quotient will be l 'a ~~ , _ T = rfi -+ CJa tf a the thud quotient being "'' ss a 4- ' = rfj ( ?a 4- 4- i j rt a 4- i , etc. etc. die the i educed uppioximatecl fuictions, the values of which uie alternately ^icvitei and smallci than the A A value of , and they appioach moie and mote closely to ,,, which may consequently be regarded as then limit The ^i eater the ntimbei of indicatois, the smaliei the difference between the approximating fi action and the exact A value of B* The following connection can be established between the fractions and the indicators :- p __ ji a 4- i) H-i 3 ^a 4- PI t on.suquently P., =3 ,, P,, 4- PI and Qj = ., O a 4- Qi- It follows, thcrefoie, as u geneial rule that P, * a, i P n -i 4- P-a and Q = a n Q M ,1 4- and this can be applied in the following mannoi : 4 2 The Calculation oj (i) Given the fi action ^ Deteimme the compound fraction, i.e. the indicators, and find an approximative fraction A _ 51 , , i = B 16 ' ^ I a, 5 3 - 3 appioached B Qi I ' * A P a s X 5 4- I 1 6 , , i i , 11 i ._ = ,..- = 7 J = . limit appro.ichc'cl still tlosci B g a 5 5 A P, 3(16) -1-3 ^i - = ^ = , = ,,, the exact valiu' B Oy 15 + 1 16' Given the fraction 399 A = 3370 __ i B 399 2 + I i i 7+6 p t ,,= , l '". 17 U j 2 {?,, = 4 < P a = , s (^^ 4- i) + <t, ~ 76 ( li - j + i ~- 9 17 y,-^= - ( , 2 i V = 7 Pj = a 4 (,,( 1 tf a + 1) +!^ + 1) = 549 Q t , t (,.^ f J) \ a a D = 6 ^ P r , = a R (4 ( fl 'i (^iss-t- + ii; t a + 0) + <*n (i<^ + 6 x 549 -I- 76 Ur, = n (4 (03 #a + i) + ^a) 4- rt^ + r 6 x 65 -i- 9 -- 399 The appioximating fractions are thus 3 j; 76 549 3370 i' 2 ' "9" ' 65 ' 399 ' Jor Screw tutting oit I,athc\ 43 (;) Deteunine the compound h action ami the appioxi- matinj; h.utmiis <>( the mmibei 2 "j 182818 28459 A 718 \Si,S2.s i $<) i = ^ j 4 - - I + p, 2 (J, ~ I - t P., ~- 2 X 1 + I - } Uu -* ' ~> \> < v l > y / >sxiii- > "=r 1 ., : X { t >' - i VJ,i X I | I > ( ) i ' Pi I I --- i P t - i x s | ^ n <Ji ~ i x H ' - ! ( > ~ I P., '<) = i P 6 ~ 1x114-8^-19 (](,=. i x .1 -M ~ 7 ( ) f) ~ 7 -()' ^' = K ' 7 (4J IMctminu the approximating ft actions for the niunbci , 7 , 7 + ,,-=? 15 P-.-333 (J.,= H)6 rt 4 t I'^SSS LU-^3 rt & 2tjJ P B 103993 (J B 33102 </ sa i ^ t =s j etc. etc, <I H = 6 44 The Calculation o/ C/iange~ll'hccls The approximating; fi actions aie, consequently, 3 22 333 ^55 i i x- j i Cl t I Hj. i 7 i 06 113 33102 Fiom which the following can be detetmmed ~~ Axiom i The diffcience between two sm cessivc apptoxi- matmg fi actions is, the signs not beiinj; taken into ronsideta- lion, equal to the unit divided by the pioduct of its or, in gcneial, v = r "- 1> " + I = ( ~ J) " * Q* ( J + , <J,(j + ," Should theic also be tlnee successive ap h actions, 1> i) |> 1 n - i * n * u \ t <}-/ Q/ <> + / the fiist will then be gieatei than the second, the seunul being smallei than the thud, etc Example (sec page 42) . A__ 3^70, B - 399 ' the approximating fi actions are 8 17 70 5/19 i' 2' 9' 65 v _ -^ H-i - 1 ; '~ 2 ' 18' 585 Axiom 2. The diffcience between the exact value of the A fi action B and one of the approximating fi actions will in- vanably be less than the unit divided by the pioduet of thts denommatois of this appioximaling A action and those follow- ing, and also less than the unit divided by the square of the /or S(rcw~tnfting on Lathe* /|5 denommaloi o( the fi act ion undei consideiation , 01, in A _ P, ( - i r A ( - i )* ( - i )" <.>,.<], <V A __ \\ i u (] < (} ((_-!- o,, ,) Application A __ 3370 H ^u() I 1 , H P, _ 17 I', ___ 76 P 4 ( ji = i ( j- ~ -' ( 2i u ( ]> 3V ~ l '7 <s ' 3')y "^ - ! ~ 599 < ~" 2 30U " ' < t (H 137 () _ J 7 < ! 399 < etc. etc 2 ^ 16 From which it follows th.it in order to obtain an appioxi mating fraction, tliflenng only a millionth pail fiom the exact vaku 1 , UK* (k'nomiiuttor must consist of at least /) figures. The tliffcrcncen between two successive api>roxiniating ft actions become ronliuually smaller, and are alternately positive and negative. The difference appionches //, and nmsequeotly the limit of the approximating fr.ution to the r A exact value of .,. 46 The Calculation o/ (. h<ui(~\\'hcci\ By mtcipolation anothei fraction can still be found behvcen two appi oxi mating h actions Gencial teim P a P 4- V 1 '*>/ " H - T l * >t - By taking in place of a, the values i, ', 3 P " _ t ), othci fi action s c<in he mleipolated between ' "" and , both of which foi m an mcicasnu; 01 diminishing thaw, as they both have the same sign (i) Required, the interpolated fractions between ' = l ^ i ^4 549 ,- ,1 f , A 3370 . and , . = > J of the fraction ,, = ' ( naie A M O, 65 U 399 Vl ^ ' ; an = 7 ^,~, = 6, 5, 4, 3, ? and i, \\ <f tt l\- t + P_, Q* ^iQ-i 4-o l' = 549 O* =- f)5 P =17 ( ) = > \\ = C> x 76 + 17 _ 473 i j . _ 3 X ;(. I- 17 \|5 () 6x9 + 2 50 () >t ~ 3 x 9 + i ;>9 P __ 5 x 76 + 17 _ 397 P,, _ n x 70 + I? ,( Mj y 5X9 + 2 ^J7 O ff ""* : x () + > *" ;jo r w = 4 X 7^ 4- 17 _ 321 Q 4x942 "" 38 (j,, i x 9 + > 1 1 The factions 93 *<*) ^45 3*1 W d7J 11* 20' 29' 38' 47* 50' u< lims " t- twccn the fi.ictions 7 and y^, which nre anpioxiniathi" 2 (J " * fractions of .- . 399 jor Strew-tutfuig on Lathe*. 47 P H (2) Requned, the interpolated fractions between ' = and . ' = of the same fraction O ! 9 IV = <i,, 1V-1 + IV ~, IV ^3 x 17 f <^59 <j tf/^-.-f <J- U "3x24-1" 7 IV =- 76 <J = 9 }} ^ 2 x 17 -4- 8 = 42 () ? x 2 + i 5 IV- ,= i7 (],-,- 2 IV = i x 17-4- <s = 25 lV- 2 == U,-. = i <> 1x24-1 3 5 o 42 3 5 consequently, the appioximutm^ fiactions -;_, " " , he between ' and _, / 3 >*J i 9 I /( ) (3) Rcquuccl, the interpolated Iraetions ' = and l j r, _ 3370 UB 399 ' ft f) = 6 #,,_,; = 5, 4, 3, 2 and i IV = 3370 U* 399 IV -i = 519 Q-i = f) 5 IV _ 5 x 549 + 7C _ ^2i IV 2 x 549 H- 7<^ _ "74 ()/-"" ex <>K 4- o 3^4 ( ) 2 x 65 -j- 9 "~ 139 *^' J * J ' S <J*' r *" J ' * " IV 4 x 549 + 7^ _ 2272 IV _ i x 5/19 4- 7^> _ r >-5 <J* ~ 4 x 65 + 9 "" 269 (J^ ~ i x 65 -H 9 " 74 IV _ 3 x 549 + 7^ = 1723 (j w , 3 x 05 4- 9 204 , . P . 625 1174 J723 2272 . 2*S21 the approximating fractions , , ' , ' ,uui thus he between / and 48 The Calculation of Change- Wheels Application Determine the compound fraction and the approximating fiactions of the numbci 2*539954, so as to 5*4 12 ' 7 er piopoiio ~ r L inch in cm 2*4 12 ' 7 obtain another piopoition as 01 ~ foi expressing the r L 100 50 1 tS A __ 2539954 = ^ B ~ io 6 3 + * 24- 1+ ' 14- ' 2+ ! I I The mdicatoi s aie consequently 2, i, i, 5. ! > 3> $, 2, 3, 2, i, i, etc. 1235 28 33 127 1049 ?22 S i i i 2 ii 13 50 413 876 The following and the approximating fiactions can he obtained by interpolation between fractions 2l and ' 2 ^ - n 50 P<J 127 Qo = 50 ^ - 3 Pr, = 33 QB= 13 "o - \ == 2 and i TJ 1 ^ 28 (J<i I I P. __a n * W I "T" J W 2 2 x 33 4- 28 =r 9<l Q* a,, Qx-, + 0,,-.,,"" 2 X 13+11 ~ 37 i x 33 4- 28 61 i x 13 4- n 24 By interpolation between s = ^3 (UK j P? _ 1049 Us 13 <Jv 413' following can be obtained for Sirew-tuttmg on Lathes, 49 IV _ a,, I 1 ,,, + \\, . = i X 127 + 3} = 160 <l a,, (} , 4-Q w . , i X 50+ 13 63 _ 2 x 127 + 33 __ 287 2 X 50+ 13 ~ H3 _, 3 x 12 7 + 33 = 4M 3 x 5 11 4- 13 163 _,} x 127 + 33 ^ 54i 4 x 50+ 13 213 __ S X 127 + 33 = ^8 5 x 5 + 1 3 -"63 __(> X 127 + 33 = 795 0x50+13 313 __ 7 X 127 -)- 33__ 022 7 x 50 + 13 ~~ 363 so that the following approximating fi actions can be found \\ , ID jo 160 287 414 541 668 795 between ^ and ^ v,, fjj , n ^ ,63' 2,3-63' 3^3 and ^. t's in Condition. (i) It is icqunotl to cut 3- 1 , thieads pet 2}J, in Leacl- icw ^ huh pitch. Fitch to he cut ~' () Lcachcicw J inch pitch Solution; 1 *' x 2 - ^^ ~ ' 13 x If - } ""x4"" 28 No wheel with ,13 teeth is to he found, and the numbci f |3 ih indivisible. It will thus be neccssmy to find an approxi- mating fraction, 'H I fi action = ( ^ -- i 4- " j 1 + 5O The Calculation of Change-Wheel^ Indicators . i, i, i, 7, 2. a, = ff a = I P (J = I X 1 + I = 2 ' F, ^ tf a = I > P a = i X 2 + I = 3 O , = I X 1 + l = '2 ( )', ~ 2 ff/i = 7 P 4 = 7 x 3 -f 2 = 23 O,, = 7x2+1 = 15 o* - *,", a rt = 2 P 6 = 2 X 23 -I- 3 = 49 Or, = 2 X i >; + 2 = 3 ' ( , 6 ~ , ^ %,,,() .' Intel polating between ^ and 1* rt w P_, -f P w n 3x23-1-3 26 , -" = " J = =- is obtained () ^O"" 1 + O"~* 1 X 15 + 2 17 28 ^ =i 5312 which is o 0045 l css tnun ^ lc Actual d.ietion J *** 26 ,. = i '5 2 94 0-0063 This cliffctcncc occuis in eveiy 2 thtcads, so that the actual diffcicncc pei pitch is <inly o 00225 appiofichcs most closely to these two, so that the wheels will consequently be 49 _. 7 X 7 _ 70 X 70^ 32 4x8 40 x Ho* (2) Requhed to cut a pitch of 3'7 mm, 1 ,cru Iscrcw 10 mm, Solution : " . 100 Theie being no wheel with 37 teeth, and the number 37 being indivisible, an approximating' fraction will have to he found. for Screw-cutting on Lathes Compound fiactum IOO 4- 2 4- 1 f 2 4- i -f Imhcdtois ,ii( v thus o, 2, i, j, 2, i, 3. -(!-,- 'Q,., ! -- ? , P. O X 2 4- I 1 Cj a = 2 ! ., i .- i i P, =- J X i 4- o = i ' O ; , = 1x24-1=3 Pi sr ^ y r 4- i j= ' x 4 "-" * f'x i ^r i ~"" . <7< ~ 2 X 3 4- i =7 i x 7 4- 3 ~ 10 P 7 -= 3 x 10 4-7 = 37 O,, = 2 X 3 + 2 = 8 (J f) =2x84-3 = 19 y (! = i x 19 -f 8 = 27 Q 7 = 3 x 27 -I- 19 = 100 10 i\ Qi l "O j Q = 2 p t = I Q.i 3 1^ J 5 4 ^^ O Q * ~ s T"^ *7 P fi ___ 10 Qe ~~ 2 7 100 The appioxiraating fmction ~ 3'74> which only chffcis * J / from the actual ft action by o*oo/{ mm. per tluead, may thus be accepted ior <ill practical pui poses, IO _ 2 X 5 _ 20 X 50 27 ~ 3 X 9 ~ 45 X 60" (j } The Proof of the Smn. The comparison thai 6*5 in, 165 mm., or an adopted fraction, is not perfectly accurate. Should it be desired to find out to what extent the fraction which has been arrived at, and, consequently, the thread to be cut, deviate, this can E 2 52 The Calculation of be done, when a mctiic thiead h.is to be cut on a l.ilho having an English Icaclscicw, by multiplying the numeiatoi ul the fi action by the pitch of the leadsciew in mm The 1 pio duct thus obtained should coincide with the' piodiut ol the denommatoi of the fraction anil the pilch to In cut, t.e numcratoi X pitch of leadsciew in mm. = denominatoi x pitch of thiead to be cut Numeiatoi, denotmnatoi and leadsciew pitch lx nu known, the pitch of the tin cad to be cut ran consequently be delei- mined On pay;e 28 the h action ^^ has In en detei mined !oi a pitch to be cut of 2 mm, and a leadseiew ol 2 tlneads pci inch The piodutt of numeiatoi and loadsuew pile h m mm. is thus 26 x 12-69975 01 26x 12 7 = 330*2. This piodmt when divided by the denominate of the d action \\ill ;i\i the pitch in mm to be cut with the* wheels deleunnu-d in, thus, 330*2 : 165 =2 ooi mm The pit< h is consequently cxutt to within O'OOI mm. 7X J3 _ 91 11X20 ^20 is given on page 29 foi a pitch of io\ mm , with a lead screw of I in. pitch 91 x 25-4 2311 'A J J ^ J ' mm. 22O 320 The pitch is theiefoic exact to within iroofjj mm. Both these differences may piactically be re^atded as of no tonse- qucncc In the case of a lathe having a metric lujulsciew on which ICnglLsh. thiead is to be cut, the denominator should be multiplied by 2 -54 The numeiator when divided by the piocluct thus obtained, gives the pitch to be cut in inches On page 24, the fraction for cutting f threads pei inch with a leaclsciew of 10 mm. pitch is mven as l(> ^ , 6x615 If the denominator be multiplied by 3*54, the* result will be 1<5s - = l6s 6x65x2-5/1 990' 6" for Screw-tutting on Lathes. 53 K.uh pitch cut is thus o- 1065656 in The exact pitch = \ in. = cri6 in, so that the thiead cut ihlfeis only by o-oootoio in Note, that when cutting motitc thiead with an English Icadsciew, the thiead cut is a fi action too coai se, whilst, on the contiaiy, when cutting English thiead with a metnc lead- sciew, the thread obtained is a fi action too hue. (/&) Fixing vp the Wheels It is not always possible to fix up the 4 wheels in the older of sequence given in the examples. FIG 10. FIG, ii The following h action may, for example, occui . 125x55' in which ca.se the wheels must be placed as per Fig. io although the wheels 30 and 55 cannot mesh. The fraction can, however, be ai ranged in another order of 54 The Calculation of Change-Wheels sequence, viz > , which makes fixmer up possible (see 55 X 125 t. i i v Fig. H), but care must be taken that the wheels of the numerator are nevci placed in the denommatoi, 01 vice vci w?. Should simple changing about of the factors in numeiator and denommatoi, 01 one of them, be impossible, the fi action is then resolved into the lowest possible factors, and .mother combination of wheels sought for, which will give the same pioportion between numciatoi and denommatoi, as, for example 30x50 _ 2x2x3x5x5x5 _ 30x40 3OX.]o 55x125" 5x5x5x5x11 ""55x100 50x110 (/) Thread-t,uttiiig wit/i Double Compound Tram. Should it be necessaiy to cut a tin cad consideiably coaiser 01 finer than that of the leadsciew, it can easily happen that the necessaiy wheels aie lacking. For example, to cut 56 tin cads pei inch, leadsciew 2 threads pei inch. The fraction is * = IO x 1 5 A wheel with 10 teeth is 56 70x120 lacking 1 . If the numeialor and denominatoi ol the fi action aic once again multiplied by 2, a wheel with i/jo teeth is obtained m the denommatoi, which us also not at hand. In such a case, the numcratoi and denommatoi of the fi action aie icsolved into 3 factors, as, foi instance 2 = 20 = 2X2X S _, 20x25x30 56 560 5x8x14" 70x75x80' Example: To cut 48 threads per Inch. Lcacksuew 2 thieads per inch. Solution. 2 = 2 2X 2X 5 20x25x30 48 480 5x8x12 60x75x80 /or Screw-cutting on Lathes 55 (m) The Cutting of Left-hand Threads . So fai, it has been implicitly taken for giantcd that only n^ht-hand tin cads had to be cut, it can, howevei, happen, though not often, that a left-hand thicad has to be cut. Foi this pmpose, the leadsciew must lotate in an opposite diicction to the lathe-spindle This is obtained by connecting up an idle wheel at will, In double ttansmission, a fifth wheel (idle), chosen at will, may also be mtioduccd A number of lathes have been constiucted of late which rend ei the connocting-up of an intermediate wheel un- necessary. With these lathes, all that is icquned is to shift the leveise-plate at the hcaclstock which reverses the move- ment of the pinions which duvc the change-wheels, thus causing these wheels and the leadsciew to lotate in an opposite direction This is a decided impiovcment, as theie is not much space to spare when five 01 six wheels are on the shear. With a double compound tiain generally the larger numbei are only small wheels, but with foui wheels, however, every proportion is possible, so that the placing of a fifth wheel can sometimes be vciy troublesome. 56 The Calt His twn c/ (Xv //<;? CliAITlCK 111 AND HiHR iONMlUH 1ION, TniSKK aic diffcient foiw. ol thread, a h-w of uhhlt aie illustiated in Fi^s. 12 15 Fig. 12 shows the Ver thiead in its "rwtal fuuu, which is conM.1 noted in different types, ,uul is inns! often met \\ith Ki^. 13 illusti.itts the stjuau' <u ll.u (Ijit'.u!, the st-aion of which is eithoi u S(|u,uc 01 a ii<;ht an;;lc, and wliit It is much in use for luis^'i dismictois and >ai',n thuads. In IM*| 14, the tiapexium tlucad is smi, Ihc Mrtin nt whu'h is a tiapc r I''J', 12, 1,3, 14, is. aium, much in vogue {or the hvuKcrcws of lallu-s, the \\uun being also a tiapeyfum thieati, FiV,, K; ,s the immd thu-ad, ioimed by the iiiiej, section of snniYuvU's, Voiy little need be said with ivft-nnut- t*t flu* last thm- types, fur which it is inipossiMc to ,spak of any om- .sys.lt'in, the form of thu section beitis; d pnnirnt tm u'n uinstanrrN and detei mined by each individual at will Different vaiiutieH, however, exist uf Ihe Vei* thrt-.ul. (/;) Tj>/>?\ f The type chiefly employed ts nnlainly the Whitworlh system ; Mg, 16 shows llus constmction, The depth of the Whit\uwlh threail t.s uqual tu u'<i,j of for Strew-tzitmi> on Lather, 57 tin pitch, ihc sides of the thtead foi mini; an an^le of 55" with top and bottom lounded tlnough \ of the line //, drawn pctpemlicnlai fioin the- apex of the tnanide to its base, tin* ladius of toundini', being equivalent to O'l (3 h Not only is ihc sectional foiui oi the Wlutwoith tlnead definitely fixed, but also the numbei of thieads pei inch foi FK. 16 h 0,96 S all dumeteis up to and including 6 inches, and this has been fixed at from 20 JJ threads per inch. The sectional loim i.s piecisely similai foi the finest as well us the coarsest threads, and it is for this icason that the exact dimensions and strength of the tlnead aie determined by the simple determination of the outside diametei. I >l.ltlUll 1 ,ll Hi) (.torn I, WHITWORTH TIIKEAD, pwmch i <> as i 1 7^ 20 ;. 7-.!^ -A! (.09 18 fl c>*<52 , '&) 7-36 1 6 I\! it'll 54 S'64 I'l i i."7o ' '39 9 <JX 13 ri '5*^7 "S 1 I2't;2 II l i ID-US , ' 15-74 It) '{ ^,"^2 '73 iH'S4 y I i<5'4 ' 'H4 21 '33 s 4 28 'W , '04 3 7 7 l| it '75 i U 7 jsG'9-j 7 DitUtu'tet at Duimetei of liuLtuin 1'hiead 111 mm in. mm >$ 34'(J2 l'i(> 2fJ'4<*> Ij 38 'I I'^ij 32 '6S *i |1'37 i -37 35-28 Vj 4 ?J j'4y V7 N *> Jo-ss i 71 *nM3 3 i 57' 15 rtjj 49-02 55 '37 f0 4S No of Tliu.itK per inch 4 4 31 70 ' 3-03 6() Ho 3| 58 T/ie Cahulatitw <>/ Chatty II 7/rt-A Table 1. mves the v.inous dimensions of tilt' Whitwoitl tin cad A Whitwoilh thiead of in tain dimensions can also hi- ui on a lonsuletabiy lai^ei otitsuU 1 diameter, the evut stu i n!' ( tl of the tin cad beim> fixed by simph del CM mining; whirl dimension of Hie VVIutuorth system is leqimed T.ible I t'.ives not only the outside- diameter, but also tlu cliametci at bottom oi llnead, so that the In ij'Jit nl the thiead can be at lived at by subtia* tin, 1 * the lattei itom the iotmei, and dividing the diffeienee by two When cutting tlueads on the latin, whu h deviate in ihumetei from this system, it is necv.saiy to know the depth of the thiead both ioi tutting msule and outside thieads The depth of the tlnead < an also be airived at by a simple calculation Foi this piu pose, just look at Ft'i^. in. By diawim; a peipendiculai horn the apex of the Uian;le t a ni'hl-antjled tuangle is formed, the smallest am;le of \\huh is equal to 55"-r2 = 27" 30'. Tuny. 27" 30' =O'^L Thciefbte, if the lunj| side of the ught-any le = 1, then the short side ^ 0*5.^, and the base tif the triangle of 55" ~ I '04 This base is, however, equal to h, i.e. the pitth. Whence it follows that h : S = i : i '<v|, or o'*)ti : L The ical depth of the tlnead is, however, only ;| li, So that the latio between the depth oi the Urn-ad aiul the pitch is equal to j h : S =r (o-yO x f t ) : i cs orj t | ; i Jj // tlm.s equals O'6/i S, If we take the outside tliametei 1), the diaitietei at the bottom of the thiead 4 and the pitch S then, d ^ 1> ** 2 x 0-64 S, or d I) i '28 S The gas thread universally adopted by the pipe tuuk*, given in Table II., Is also according to the Whilwtnth sysluin, and in 1903 was also adopted ua the .standard thread for pIptjH and fittings for gas, water, and steam by the AsstMUtitlon of German Engineers, the Assocititioti of (icnnan Fltimber^ the Association of the German Central Heating Industry, and the Union of German Pipe Manufacturers, for Scmv-tutluig on Lather 59 On the othct hand, in the autumn of 1898, an attempt was made by ,i niuulu'i of mllnential associations oi Continental en<>iiH'eis, assembled in confess at Zuiich, and including, amongst nthets, the Association of Gennan Kn^meois, the 1 \III I II \VlIll\\URII! SlUlWlNl. Tllkl',Al>. 1' vti_tn il in i nitu ! HI <mn in mm in nun in h } 17! ^j u 71 ^( 8'^ 28 lA }8 ) l 882! 47 81 I 7<>S 44' J j o 35 -qi8i} i->j hi u 41 "9 ill 1i'-7 ' u *' i S 1 i3 r 904 48 li ' 9 S' '<)S(t't(> f>7' '$8iji4 9i 19 14 44 <|S- f H7 S 2 J 9} 49 S.'d jj '15 H7J 9<>"-'*9'i 8 1 20'' ,J 'it, os'f oj. lad 4l' '<).19 '( 14 2 50 8 U vi 7 59 f>i- 33 5f> 6 ^ u 1 1 2} s?* 1 ^- 3 5'^7 f'S 7- 2 "l7 i ()2 7^ n H 33 03 5 '> 7^'-'^ S8a 73 =7 ?, 2.-".v t uS<HO J 1-1)9727-87 14 .',' (ly-K i 2tj'J t ^iKjhiV4 t ' 192 i. 'H ii 3 762 J47 247i M 4*5 ^8 SM 3 (lS 912 99 V'3 795 |3 75;* (J 5 -rjaj i 74$ 41-91 1-^338-95 H 4 101 (i 4339100-2 4223 4 1/32 i -028 41 '36 II 79 5 1 u 8S 5i II 9 f ' 39 i 1 [07" 26 II AIII.K III. ~S. 1 Ti ii.xm 1'itih l)i imttu at lUiHnill 1 Main Ptlfh 1 ll.Ulll'U I ui 1 hicail ))i mi 1'itrli Duuuctii nl Tluuwl mm mm nun iinu null mm. nun mm nun > 1 V7 20 2-5 if)' 75 48 S 41 ' 5 7 I > 7 22 z'S 18 -75 52 5 45 '5 8 i *a^ ^'37 24 ^ 2()'I 56 S'S ^ ^S 9 i j<; 7 37 37 ? -53 'J 60 S'S 57-85 10 rso H'o<$ v 3'S 25*4S 64 6 56 c)i> H i*9> y>5 3^ 3 S 28-4^ 68 6 f)0*02 U* i 7S 9*72 30 4 30*8 72 f>" 5 63 '55 M 2 IP4 39 4 33 ' 76 (>'$ 67'5S if) 2 I3'4 42 *l S 36-15 80 7 70-09 8 'S M'75 45 4S 39 'S 6o The Calculation of Change-Wheels Swiss -Association of Machine-Tool Makcis, the Society the Encomagemcnt of National Industries, etc., to replace Whitwoith system, which is based on the English system measurements, by a metnc thicad, and it was unanimously decided to adopt the S I thread (" Systeme International ") ' as per Table III Owing to the universal application of the Whitworth thread, the innovation makes but little headway, thong" I" 1 especially of late years, this system is being more and used on the Continent, especially by the Automobile for thieads cut on the lathe. FIG 17 The construction and form of the S I thread is mvera in. o Figs 17 and 18. The apex is an angle of 6o n . The section is consequently an equilateial triangle Hence it follows that the peipendicuhu h, dropped from the apex to the base, is equivalent to The truncation equals | /t t so that the thiead has a of 0-75 A, 01 o 6495 S. for Screw-cutting on Lathes 61 Whilst the Whilwoith tin cad beais not only at the sides but also at the bottom, the S I thiead, on the contiary, has a play at the bottom of, at the most, ,',, //, equivalent to the half tiuncation, the rounding of the thiead is equal to the A/vv : \/v:.;n \, , D D, \ A Y ' ^/ ,* t FIG 18 play, the radius of the loundmg in this case being j l () // The rounding and play amount, as is generally accepted, to at least ^ /{, Loewe stakes ,in aveiage foi this, and fixes the* play and loundmg at ^,, h The outside diametei of the male-si lew is thus smaller than the diameter at bottom of the thread in the nut, and FKJ, 19, vice versA, the diametei at bottom of the thread of the male- screw is smallei than the outside diameter of tin cad in the nut. 62 The Calculation of Change- Whech If we take the play a, then the actual depth of the tin cad of both male-sciew and nut equals o 75 // + u If we fix the play at its maximum, equals T \, k, then the height equals 0-0625 /; + 0-75/2 = 0-8125 /*, 01 o 703625 S = ~0'7S. The Lowenheiz thread (Table IV) is in j>onei,il use up to fc. IV Lowi-Nnii.it/ THREAD dim Pitch Diimetei at Bottom uf Thread Diam Pitch Dmuctu it llolttiin uf I In cad Diiin 1'itih Ui unitu it Bottom Of IllK.ul mm mm mm mm mm mm nuu mill HUH I o 25 o 625 2 6 O' '45 I 925 S'S y 4 15 I 2 o 25 o 825 3 O' '5 2 25 6 i 4 S I 4 o 3 o 95 3 5 o 6 2 6 7 i i "5 35 I 7 o 35 i 175 4 o 7 2 95 8 J 2 6 '2 2 o 4 1-4 4 5 o 75 3 375 9 i * 7 <>5 2 3 o 4 i 7 5 o 8 3' 8 10 i 4 7 9 a diametei of IO mm foi instiuments of eveiy description, especially in Gei many and Swit/eiland, and in scicw works, the sciews are almost exclusively made by this system The construction of the Lowenhci/ thie.ul is shown in Fig 19. The apex is 53 8' This angle icsults from h ~ S. The tin cad is titmcated flat on the outside diameter and at bottom with a J truncation, so that the ical depth of the thread is = 0*75 //. The Sellers thiead (Table V.) is an Ameiiean thread, con- stiucted as pei Fig 20 for Strew-cutting on Lathe*, TAUI v, V - Sw i FRS THREAD. Nunilii r >{ I'll! < U.I' )ll 1 UK ll Nmubtrof I'liu ul pen mi h 1,1 Mini ter Ntuuhii ol Hiu id, pu inch .JO 24 20 IS 16 M M ta JX 1C) 5 S -ll 4 4 31 31 3i 4 4l 5 Si SI si 6 3} 3l 3 3 The apex is t in an^le of 60, so thai the peipcndiculai f { (hopped fu)in the apex to the base, is again = o*866S, The tlueacl is Hat-faced at bottom and on the top with J trunca- tion, consc(|tiently t = ,"] fj and 0'7S X 0'866 = O"6495 S. The thread which resembles the S. I. tin cad veiy much has, however, no play and is divided accoiding to English measuiements. Although largely displaced by the Sellers thread, theshaip V thiCiid still exists and is used in America, (See Table VI.) The section is an equilateral triangle not truncated. 6 4 The Calculation of Change-Whccl\ The B A. S (Bntish Association Standaul), as per '1 able VII, is an English tluead much used in Kn^land foi sne\\sof small diamctci, especially foi electiu fittings. Tin 1 - a|H \ t Fig 21, is an angle of 47 \ n The thiead is tninratt'd, and ' VI SUAKl 1 V I'lIKI AD Number of Number ot Niiuiln i of j Niimlii i nj Dnnit-tcr rineads Dmiitlc.1 I liu uU Diumui Iliiivid hiuiiilii J'iin ul , ptr mi h pi.r UK h inch inch mi h i 20 Hi 10 n 1 ia IS I o i/t ,! 16 1 fi l ii 9 > I 7 i\ 14 I ,S * H i 12 Ik 7 2 ! A 12 1} 7 2{{ 8 II I ft' 6 2\ 3 1 II ii (> ^ i IO 1 5 "5 j) top and bottom aie rounded, leaving the depth of the equal to o 6 S In addition to the foi cooing, the Delisle, Sauvnje, Acmt 1 , and Thuiy systems aie to be met with. The total number of thiead systems exceeds fifty, hui only the seven most used have been treated of ht'ic, TABLE VII B.A.S. THUKAD. Number ta l r m etei> 6 5'347 4'i 3'6 4 va 2 H 2'S '. i-Q 1-7 r 3 i ^ i 0-90 810 730-660 Spo'S^o^Ko'o'io-.- Jot" Xi on Lathes, ((} Si i ew-c lifting Tools. A tool used ioi sucw-cutlino m iist fust and foiemost be peifet tly tine. It is not to he looked upon as an oidtnaiy tool, noi may it be mound on a stone which does- not mn line When tuttim; deep threads, whethei they be V 01 squaie, it is always advisable to use sepaiute tools loi toughing and hmshim, The cutting an{le must be about 70, whilst the tool must not !r pointed 01 somi-t nculai, but flattened at the cclqc (\ f li\\ n 2 and ?3), as othenvise the angle will not be true, and, .it the same tune, it will be impossible to nrind the tool accurately The tool must not only stand on its edge in the alible B, IMJ,' 23, but the sides A A must also have cleaiance, The atisde in which the thread lies on the work has also to be lakvn into etmsidetation, and the line AB, Fig, 24, must tun .& Fie.. 2<j, Fir., 26. at the same angle. Suppose that a I in. pitch has tu be cut on n diameter of 2 in. Then, imagine C D in Fi^, 26, to be the angle a I which the thread lies on the woik, the line AJBof F 66 The Calculation of Change II '/r<-A the tool, Fig. 25, must thus run patallel to the line (' 1) in Fig 26 This is still moic evident in the i ase of sqnaie thi^cads with a coa ise pitch, Fig ?7 In this i ase, tlu <kuan<e on the sides of the tool must be diOcient, The diametei of the tin cad on the top, as ,ilso the angle of the tluead thete is indicated in Fig. 28, that at bottom of the tluead in .^ Fig 29, a and b being the ehrumferenee,, t and c the pitch, which is tlu* same for both, and thcic aie consequently two angles* The hypotenuses if/ and <* show the un^lc of the tin cad til top and bottom. If the cleaianceol the tool is tont'ct on the top.it will be mcoueet when at the bottom. The hteepei the pitch, the moie noticeable this will be. The tool must have more elearanr e on the right-baud side for bottom than ut the top, but less on the left-hand side, The tool must consequently be ground in such a manner that the tight hand side will have enough elearauu' at bottom of the thread, whilst the eleanmre foi the loft- hand side must conr.iu with the angle ul the top, that is to say, foi a light-hand thread, as in Fig. i*7 ; for left-hand tin cads 01 for internal threads the opposite conditions will exist in icgard to angles. The tool must menrdingly be. It C-i FIGS. 28 AND 29, pound as indicated in Pu,> 27, A R being- the slope of the iKjht-hand side ot the tool, A C on the left-hand side. The iippi-i cutting smfart' of the tool must um squat e on the line A l>. When <uttim> an inside linht-lund thiead, ey;ciythinj> is i excised, wh.it is li^ht-hamled becoming left. Foi a Vee tht cad, the tool must be <round in accoulance \\ith the aU",lc- of the system ol the Ihioatl, ft need scniccly be said that this must not be left only to eye 01 the lough estimate of the opeiatot, A tauge should be piovided, as V'W'I'r'i!^ pel 1'V, 30, ;ivnv', the pierist* un^le. And yet, notwithstand- ing that it is lai nou k diHieult foi a woikman to judge an tun;/! 1 \uth llu e)t- thun to ^uess i ceitain /etigt/i, and no one would eun think of peimittin^ an opeiator to estimate a u'lt.un length without usinu his rule, it is- an exception when the opnatot K ptovided with n suitable an'le gauge. // /\ nttt'i'ty iw/xnA't/'fc that a thread can be trite when llie tfuttttH JIM juried //!<' rW/i' J the ttwl with his naked eye. This {juuuc purpose 'kvrn FIG, a .second, and not less important, the tool be gnound to the precise It is htill possible to out a wrofl^ thread, for the tool must be M pkeed in the holder that an imaginary line diawn pinpcndteularly tnn the apex of the triangle to the imaginary ImstMUUht alho fall perpendicularly on the side of the cylinder on which the thread is to be cut, Not having this gaugc, I? 2 68 The Calculation of Change-Wheels the opeiator judges with his eye the position m thinks the tool should be placed But the most woikman can make a mistake, it is not possible ^f to be ti nc if the tool has been placed with the in the position which might icasonably be collect, and this is afteiwaicls checked with an FIG 32 it will almost invaiiably be found that the position r The icason is that the two lines forming the ang>-lc . vciy shoit in piopoition to the othei lines of the t:o< being consequently deceived In Fig. 31, at A, Is shown the manner of gauj^inj to which a lathe centic should be tinned, at B, tl FIG. 33 which a screw thi cad- cutting tool should be C, the concctness of the angle of a scicw thieacl -<\ In Fig 32, the shaft with a sciew thiead is ^vip held between the ccntics of a lathe. By apply In as shown at D or E, the tin cad tool can be set zxt: for Sircw-cnttwg on Lather, 69 !<> the sh.ilt, and then fastened in place by the boltb m the tool post, tlu-U'by avoiding nnpeifect 01 leaning thieacls ' n I'" 1 ", 33* <-t V and G, the mannei of setting the tool for < lilting internal tin cads is illustrated (d] Cutting t/ic Thread As pievumsly stated, it is always advisable to begin cutting a IhuMcl th.it has anything like a deep curve with a loughing tool which us at a cutting point and which need not be {Ljiound piecisely to the angle. The thiead should afteiwaitls be gone ovei with a finishing- tool When engaged in cutting shallow threads, the tool can cut on both sides at the same time, and it can be put exactly on the duection of the shaft. With deeper tin cads, i.e. quick pitches, this is no longei possible. Cutting with both bides of the tool at the same time causes it to snap, the thiead is tough, and veiy often it is impossible to continue working; the tool should, theiefoie, woik but one side at a time, should fiequently be set slightly m a paiallel duection to that of the shaft, and cliiectly theie is any play between the tool and the thread, it must again be set squat e on the direction of the shaft. Each time that the tool has gone completely ovei the thread, it should be withdrawn and again set in the oiignul position at the commencement, though increased with the amount cut at one passage For this piupose a gi actuated collar is provided to the feed screw by means of which the traverse movement can be read, and by which the tool can be set in the exact position every time. "The opciator foi racily got out of the difficulty by maiking the position of the screw spindle with a piece of chalk, On lathes of up-to-date consti notion, the giaduated collar is now always to be found on the sciew spindle. A very practical consti uction is shown in Fig. 34- Advantage is hue taken ol the movement of the two half- nuts when opening and closing, to withdiaw the cutting tool irom the cuive, and mis vcrsd, back again to the exact The CaknlafKni of C/ian^e-ll position, so that instead of having to cany out vaiious opeia tions at the end of the thtead, a simple movement of a handle is till that is icquiied. The constiuction is as follows Ovn tin* (\\o half mils which move unclci the carnage in the .same du tit ion as tin cioss-slide, and aie opened and. closed by a double n;;hl- aiul left-hand sciew, is placed a Pi-shaped slide lived mi knobs of the upper poition of the half-nuts, 'I he su< w spindle ol the cioss-slidc fits in the uppci poition of this slide on the one side by a turned up ed^e, and on the othei by huk-nuts The scicw spindle must consequently follow the movement o! the slide Holes aie dulled ii ( idU thtoii",h the ptojetthu 1 , / v r , ,, * ' I I "\ j- -^,J,ir""r J^' ",, * j f ^ r U ^) paits of the half-nuts, and the slide, A steel pin lit-, t fnsrly into these holes. Oblong holes, in which the pin has play, air boied in the carnage for same. Before beginning to cut internal or evtenul llm-suK, tlu* pm is set in the foiemost 01 hindmost nut, so that the halt nut thiough which the pin is placed Ls coupled with the slide In which the scicw spindle fits, and umseqiiemly they imtsl follow the movement of the half-nut in que.stion lont-ther with the cross-slide and tool. It is wotked as Jollmvs: As soon as the tool has anived at the end ol tin- threat!, the half-nuts of the Icacl-scicw are opened and by this means the tool is withdiawn fiom the thread The caniagu j then liftin on 71 n tinned l>\ hand hy means of the pinum, the tool set so imuh faithet in with the sue\v spindle as it is desiiod to cut drepej, and tin- half-nuts aie closed again This causes the tool to lesnnie Us oiigmal position, only cutting the uiatcnal st much deeper as it has been set Luther in by hand. If no thtead is lo 1> ( nit, the connection between the slide and hall- nut is Inoken by withdrawing the locking-pin, and the slide is coupled to the uoss-shde by insetting the pin in the hole boied tlmniyh both slides When screw cutting, this anangeinent ies.ults in a decided I saving of tune, l>esiilcs preventing the possibility of mistakes arising from inserting the tool eithei too far or not far enough In, There should be an outlet for the tool at the end of the 72 The Calculation o/ C/MH^C l\ f Jn.cL\ thiead If the diametei is sufficiently lai;>e to pet mil of it, an entiie circuhu gioove should be tinned, Ki<> ^5 If, ioj some reason 01 othci, a citcul.u gtoove is not possible, a suitable outlet, as pel Figs 36 39, must be dulled foi vee ot squaie thieads Bcfoic commencing cutlin<j ( , the tool should be so fixed that it will antvc just at these holes. It was foimeily the custom to tetuin. the uuna^e when the tool had gone over the tlne<id, by tevoishi", the movu- rhent of the lathe But with the ptesenl-day eonsiuu tion of the lathe, by which it is possible to letnin the carnage quickly by hand by means of the handle, the h, Ul-nuts ate opened and the caniage leturned by hand. If the thiead bcttu; cut is of the same pitch as, 01 an aliquot pait of the pilch of the Icadsciew, the half-nuts can be chopped into engagement at any point of the Icadsciew without any difficulty, the tool always ictuimng to its pi ease position in the thiead This is, however, not so when the nmnbei of thiead s pet inch aie uneven 01 bioken, and othei means must be adopted to ensnie the tool ictuintng to its precise position in the gioove. Con- sequently, when slatting to cut the Ihtc.ul, a slop, ot mm king line, is placed on the bed, the half-nut closed and a ehalk line diawn on top of the leadsctew, and anothu i hulk line nt the fiont side of the chuck-plate. When the tool bus none oyet the thiead and the can tug c has been teUnned by hand as Jar as the stop or the line, the head spindle is turned lound till both chalk lines are again in then original position, the nuts closed, and the tool is once moie m its pieuse place in the path which has, just been cut, This compaiatively troublesome and punnUve manner of working is done away with, if the c;iniaf;e is piovklcd with a thiead indicator as shown in I ( "i^ 40, The following is the principle of this uttat hment ; A stwvll worm-wheel runs on the leadBeie\\ and by moans ol a pinion gcunng, causes an indicator to move on a (iicular indox- plate. All that in now nuees-saiy is in iuli* the position of the indicator at the starting point, allot which, tlu: half. nuts can be closed, mid the tool will come pieeisely in the path each time the indicator resumes its original position, for Screw-cutting on Lather 73 (?) The Cutting of Double or Multiple Threaded Screws The cutting of double 01 multiple threaded sciews causes <i ^<>od deal of tioublc, as, in addition to cxeicisiny ouhnaiy uue that the thicud cut is tine, anothct most impoitant point has to be taken into consideration, viz that the setting of the tool is also exactly equidistant The manner of working is smiilai to that foi a single thiead, but caie should be taken as fat as possible that when cutting a double thiead the spindle wheel is divisible by two, and foi a ticble thiead by three After the fust incision has been made to the lequned depth, the tool must be shifted exactly to the centie between Fltr 40 two threads for a double, and to one-thud of the intei mediate space for, a treble thiead The distance the tool is to be shifted should, however, nevei be mcasuied off, as this can novel be exact, but must be obtained by mechanical means, either by turning the woik-piece while the Icadscicw is slationaiy, or by turning the leadsciew while the work-piece remains stationaiy, If a double thiead has to be cut, one of the teeth of the spindle wheel coming between two teeth is maiked with chalk, as also the two teeth which the tooth in question engages., After this the spindle wheel is bisected and this tooth is also chalked ; the spindle wheel is then icleased 74 The Calculation of Chaugc-Wltceh from the wheel it engages, the spindle is given half a turn by hand, so that the opposite tooth comes between the two maiked teeth, and the two wheels rue once moie engaged If the spindle wheel is not divisible by two, then this must be found on the wheel on the leadsciew, but the pitch of the thiead to be cut must in this case be taken into consideiation Foi example A double thieaded sciew of 4 thieads pel 3 inches is to be cut on a lathe with a leadsciew of 2 thicddfc. per inch. The fi action is -i-= 3 = 7 *> . \ ^ 5 The spindle wheel is, howevei, not divisible- by 2, and at. the factor 3, which is indivisible by 2, will invanably be found in that wheel, 4 wheels aie used so that the factot 3 can be placed in the intei mediate wheel 75 _ 100 X 60 50 ~ 50 x Bo If there is any reason, for instance, with heavy lathes not to turn the spindle but to shift the carnage by turning the lead- screw, this is accomplished as follows foi the above example Pitch = | in The carnage must thus be shifted ,"/ -f- 2 = | in , the leadsciew has a pitch of ^ in., and so must make 3 1 = 1 revolution, the wheel of the leadsciew has So teeth, and consequently 80X7? 60 teeth must be moved If the same pitch is to be cut on this lathe but foi a thtee- thread, then the first-mentioned wheels, ?5 arc the best to SO use , the wheel with 75 teeth can be divided into thiee, and 25 teeth turned each time. If it is desired to move the caniuge, this must be moved T-T- 3 = i i"-, the leadsciew make 4 i evolution, and the wheel with 50 teeth be moved 50 X J => 25 teeth. For example, To cut a pitch of ij in. Double thieaded screw Leadscrew I in. pitch. Solution IL = J 5 = I0 x 6u I 8 40 X 80 " jor Sit'ciu-cut/iug on Lathes. 75 Poi ,i double Ihieaded sciew, the spindle wheel is divisible by \ \\ -i- -3 = J~; in Thi.' lead sci ew must thus make { (j ~- l } [\ revolution, |[| x Hi) = 75, The wheel on the leadseicw must thus be moved 75 teeth, K \*amphv~ To cut 6 thieatls per 15 in, thiee-thieadcd suew, Leadsuew | inch pitch. c i , If' 15X2 75 X 80 Solution ' JUi_=: > = ' J \ () 30 x <\o Poi a thiee-thie.ul, the spindle wheel can be divided into 3 X 25 teeth The Cciniagc must be shifted \, 5 3 = il m ' so thal thc leuclscicw must make ,] -f- J = V' i evolutions The wheel with 30 teeth is placed on the leadscicw, and 30 x \l } - 50 teeth ate moved = 50 -r 30 = J i evolution and 20 locth. (/) 77/6 1 Cutting of vet y Coarse Thread. When cutting coarse tin cad, a difficulty may possibly occur which will lequire cuicful consideration When the tin cad to be cut is considerably codiser than that of the lead- set ew, the movement of the leadsciew must be appreciably quickened, There is, however, a limit to this, and that is the H-sistante olfeied by the teeth of the goat -wheels If the pitch is too coarse, these will bieak off The extent to which the pitch may be mutased depends, rmtmally, entnely on the stien^th of the wheels supplied with the lathe, Generally speaking the pitch may .safely be a four-fold of the leadserew, anything exceeding this being attended with considerable danger and the off-chance of the teeth breaking. hi order to permit thread to be cut which is many times toaiser than that of the leadscrew, a gearing can be attached to the fast headstoek, as illustiated in Fig. 2. The wheel 15 can be set in connection with the small geai- whuel of the double back gearing. If then the lathe mns with 7 6 The Calculation "/ Change- 1 ! 7/< , A double back gea,, the ..ilu, "I -P-l l-euveen UH ,..,- Pulley and lathe spm.llc w,ll lx- I K, 'll IK-IWC-,,. .lu- .-1 . , ... ul the cone-pulley pm.on 2 ,, ,u,,l the u'h,, 1 ,u w,l U,,,,,,,!, U 4 .evolutions l<> i of Ilu- sp,,,.IU- Inllu-.-.s,.,,!., ll,,,...,l vvh.ch ,s foiu times auisci th..n the l,.,Kuew, Iheu ,, .1 ,.it,.. of I ; i between the t-han^t; \vlnvis, whil ,( tui ,i ihir.ui i-s^JH times as coaise, Lheie is only a ratio oi i; TaUm", ,f uii\i* th.it the teeth die stump, etioii-'Ji UH a talus ut I ; ,|, aiul that the leadsctew has i in pitiii, then u \ x.j X | K HI, pit'h may easily be cut in this l lor Si moulting on Lathe\. 77 One of the in \\i-st designs fot sc rew~cuttm^ is that of the litMhl, v-Muth.n Msicm, \\hieh, ), y HUMUS O f attain of ocais p!u<vd umlei ,md ,U the side oi the hcrulslock, icndeis it l" IN '' !blr to (Ul 'i nuinbei o| thio.icls o( thfToicnl pitches )s ~v*;,'v.v {'"' ^'^^(tWMJSSff MS f /////$/>$, , i ' ' l V '\NN\\V. '11 * v-if^jMc "V 1 i *W H 'TfT ... ! ' ^ ' ' i! i! ^ !-i irn FfC 42 with* nit the iHTt'ssity of fixing different ch<mge~wheels. ("halloo wheels as they have up till now been imdei stood JH t'nnnv{}tii with the htthe, luive been entirely supuiHcdcd On u Ltlht 1 piuvided with the llendey-Norton system, it is no lon;;<M ju'tt'ssary to fix up or lake off change-wheels, the various whooK being .simply and solely geared tip in the space 78 The Calculation of Change /77/6rA formed between the spindle and the leadsciew by the shifting of handles The calculation of change-wheels is consequently a thing of the past But, in this woik which tieats of the whole question of screw-cutting in an nbndi>ed foim, a descitption of this system, which will ceitamly come moie and inoie to the fmnt in the stiu<lcfoi economical tools, and has ahead} been vciy largely adopted, must not be missing " 4 " , VL *> m^'T^s .- /A X^m^i^ \ ', ;,'?:' . \ ^ ** Aruuigemenl o{ wheels m a L<xl|;i> anil Shipli y hulu, the dutt On a lathe of this description, scicw-t uttin^ has Ix-cn i educed to its simplest possible foim, A drver \vtu*Kaitn may, it is tine, be quite capable of e,duil,itin } ", the \vlu k cis leqmred to cut a certain tin cud quirkly, and cun possibly reckon it out m his head, but even so, llu- utlual fixing up of the wheels seriously interferes with the strady pu>^rcs of for Screw-cutting on Lathes. 79 the woik, whilst the difficulty is at once doubled whenevci turning, chilling, and thiead-cutting have to be pcifoimed periodically, as, with so many lathes, the attendant chcum- stanccs ,ue such that it cannot be aridiigccl foi all at the same time The lathes undci discussion ate constituted in such a manner that a gieat vancty of threads can be cut without icquuing the fixing up 01 taking off of a single wheel In the eailici constiuctions of this type of lathe, theic was mvanably one gieat drawback, viz. that the numbei of pitches which could be cut was comparatively small (10-12 pitches), but this numbei has now been extended to fiom 40 to 44 diffeient pitches. The foicgomg illustiation (Fig. 43) shows the complete anangemcnt of the wheels This gives a cleai view of the bed, the fast Deadstock having been icmovcd foi the piuposc The anangement of the wheels consists of two sepaiate groups of wheels The fust gioup (9-11 wheels) is placed undei the headstock, the second being in a closed box attached to one side of the lathe The action peifoimed by a wotkman in geanng up the wheels foi the cutting of diffeient pitches is cxtiemely simple, so that aftci a biicf explanation it is sufficiently cleai even to a novice, tmd it can be executed so quickly that not more than fiom 10-20 seconds aie icquiicd to change the wheels foi anothci pitch than that foi which they weie geared up. An index plate is affixed to the gear-box, which is given on page Bo In its exact si/e. A handle with pointer is placed undci the plate. This pointei can be moved over the cntue length of the index plate and set in the middle of cithci of the foui divisions of the plate. This handle is connected with the wheel indicated in Fig. 41, by the number 862, winch accoidingly moves this wheel with it , whilst nuclei the holes in the headstock the numbers I 8 or i~ii appeal, accoidmg to the dimensions of the lathe. If, lor example, it is leqmred to cut $2 thieads pei inch 8o The Calculation oj Changc-irhcch Thds Knob Thds. Knob I lid', 'I lid KlHl 18 19 20 22 23 24 26 28 3 32 3 4 5 6 7 8 9 10 ii 9 <)1 10 ii ni IS 13 M IS 1 6 3 4 5 6 7 8 <) i<> ir I 1 J, 1 S s 1 Si 6 (' 7 74 16 3 I S (. 7 S It) 1 i (I 7 S <) lo I J 80 to 40 40t() 20 Pi A I I? It) to the pointci is placed by means of the handle in the middle of that division in which the numbei in question appeals mulct the letteis Thds. (Tlneatls), in this paitieulai ease, in lh" second division on the light hand '-ide. On the same line on which the number 5] appears, the fignte f t 1 ? to be found. The handle on the head stock is now plated m the hole above the figure 6, and the wheels me than <>eaied up foi euttitif* the desired thicad For all other thteads appearing on the index plate, the ptocedme is identical The topmost handle 957 ls pl acc d m the highest 01 lowest position, uivoidhi^ as it is desired to cut left or light hand thread We will now pioceed to give a detailed description of the constmction of this gcanng. Wheel 968 (sec Fig. /|2) is fast on the lathe, spindle and engages wheel 922 (Fig, 41) whunevet right -hand thread is to becut In this case wheel 933 is idle. Kot , i left-hand threat!, wheel 968 engages 923, and wheel <).L' i.s caused to total e by wheel 923, so that the direction of movement is just the rcveise to that in the fust case. Both wheels run loose on studs fastened in plate 920, and arc- shifted by the middle for Screw-tutting on Lathe*,. 81 handle Wheel 922 engages wheel 955 which Is fixed on shalt 992, which is consequently biought into motion, This same shalt ^52 nnpaits motion to wheel 959, which, by means of a key way, can be moved in a tiansvcise diicction by the handle undei the fast headstock Wheel 959 engages 961, which can be geaied up, by means of the handle ahcady icieued to, with all the diiletent wheels 651-659 mulei the fast headstoek, which wheels aie all fixed on shaft 662 , wheel 961 consequently impailhig motion to the shaft. Wheels 666 and 067 ate also kej ed to shaft 662. Wheel 862 (Fig 41) movable by a keyw.iy, is mounted on the leadscrew. Con- sequently the motion of blufl 662, to 'which the gem-wheels ate keyed, is ttansmitted to wheel 862 by one of the wheels 606 01 (167, vid two sets of double wheels 905 and 906, both of which setsaie identical This tiani of geais can be seen in the detailed drawing* Ktj- /jl, to the lelt ol the side view of the fast headstock. Jt should be noted that wheels 905 and 906 aie coupled, but that each set is independent of the othei, and can consequently lutate at diffeient speeds , this is, moieovei, appaient with the whole turn of geais, seeing that, whilst wheels 666 and 667 also coupled, and each engages one of the sets 905 and 906, the latter obtain vaiious speeds. This tiam of geais gives four diffeient speeds between shaft 662 and the leadsaew Wheel 606 engages 905 and 906 on the right Wheel 667 engages 005 and 906 on the left, By moving wheel 862 on the leadsciew (this wheel is also to be seen in the illustration, Fig. 43), and by changing handle ydj, which tuins on shaft 662 and to which at the same time the two sots of wheels 905 and 906 are keyed, wheel Kfo can be placed in lorn different positions, i, 2, 3, and 4, (See detailed dt awing, Fig. 41.) Wheel {>(; =- 906 and wheel 666 = 905 =s 862. The proportion of 007 to 906 .= r I, of 66(>to 906 =2 i, so that If wheel %j engages 905 on the light, the speed of shaft 662 * ^ V* ^ i,s doubled, seeing that ** 2, ** i X2 If wheel B63 engages 906 tm the right, the motion ol the G The GittM/tithw of C/Miigt- //7v'A shaft is tiansmitted without nn)' vanation, am I ulu-t i <,<< n the light simply serves as an idle wheel, li ,H* > t n-; ,t",e<, 'in;, to the left, tlieie is a double i eduction in sped, U ,Sn* < n^a-M 906 on the left, the diminution is lour tunes is 'iieat. <*tm scquently, if the handle on the fast headsttu k is set t <j nut" No 9: With the pmnti'i in uihmw J, ^ iiUh<', pt-t n It *'. '/' ). *4 4, 2K will tn nt In this manner, with n wheels on shaft fh\ 4 j tltllei pitches can be cut. The swing plate of the f.ist headstoek IN hutlun * tun structed that, by netting up oius wheel, the spt'ttdu! tlt< lt%w!- scicw can once more be doubled, or by removing fin* ?..wu wheel, it can be reduced to half as slow ugahi, so that all flu threads appearing on the index table can now be ujl, with double or half the number per inch, The reseive hle tn the swing plate can be cleaily .seen in 1%, 41, ulM* to i,>,*j, In the foic^oing illnstraticniH, Fig, 44 glvc;H the ebljit- lion foi fine threads, Kig, 45 for coarse threads whilst W|t, *|6 shows the position of the wheel 955, for St rew~i ntting on Lathes. 83 Tin- iisu.U ;e.utm Is: Wheel 968 engages 922, and 922 ur;.tn's u^, the wheel on shaft 9152. For fine thieads, 968 \ U'si'tes OJ~, and c> >2 engages o?3, consequently 923 engages t> {, n'luch IN ,i double whet I with 9?$, the proportion between them being i 2, Finally, 923 engages 955 Wheel 955 d*H% nut engage <)2\ hut is moved a wheel's width to one side (See Fig, 4] 6) MfH^iAft TffltCtOS Of /NOKK Frit. 46. Km rtmisi^ threads, qf>8 engages 922, and 92 i.tiu't|iwntiy ^1 imij.mt's 923 and 923 engages 9^5. Fiom ,i t',m*lut ionsith-ration tf these two comb!natkm& foi fine and MMi.r tlsie.nN, it will he seen that wheels 924 and 925 on tiu* OIH- suit;, atul wheel 923 on the other side, are mutually iftfti*h.ini',*Hi foi the two cases. Sfai It !WH ontyhocn multiplexor fractions of an inch, in both, which nmia !>o cut in thin manner. Should it, how- rvu- ht Ht'WN-ary to deviate boredom for any special pitch, tlwi thrr,tits than ihoscof the KnjUHli system can also be ,-irt by 4 u-ruin proportion between the two wheels 924 and 0-5. 3992 /i